bio-inspireddynamicradioaccessincognitive...

Bio-Inspired Dynamic Radio Access in Cognitive

Networks based on Social Foraging Swarms

Paolo Di Lorenzo

Universita degli studi di Roma “Sapienza”

Dottorato di Ricerca in Ingegneria dell’Informazione e della Comunicazione

24 Ciclo

Advisor: Professor Sergio Barbarossa

May 8, 2012

ai miei genitori,

Abstract

There is strong trend, in current research on communication and sensor net-

works, to study selforganizing, self-healing systems. This poses great challenges

to the research on decentralized systems, but at the same offers great potentials

for future developments, especially in view of the current trend towards minia-

turized systems. Even if the development of self-organizing systems is probably

at the beginning, biological systems offers many examples of self-organization

and selfhealing. This is as testified, for example, by swarming behaviors, brain

activity, and so on. It is then of great interest to derive mathematical models of

biological systems and see how they can suggest novel design tools for engineers.

Signal Processing can play a big role in this cross-fertilization, as it can help

to find out manageable mathematical problems, study their behavior and test

the performance in the presence of disturbances. The challenge is to establish a

cross-fertilization of ideas from biological to artificial systems, as well as to help

understanding biological systems as such.

This dissertation considers the problem of dynamic radio access based on sensing

in cognitive radio networks. In particular, we follow a rather alternative path

with respect to more conventional approaches and, inspired by biological models,

we formulate the search for radio resources, i.e. time and frequency slots, as the

search for food by a flock of birds swarming in a cooperative manner, but with-

out any centralized control. The interference distribution in the time-frequency

plane takes the role of the food spatial distribution: The birds (radio nodes) fly

3

(allocate their resources) over the regions (time-frequency domain) where there

is more food (less interference). During the flight, the birds move (choose their

time-frequency slots) in a coordinated way, even in the absence of any central

control, in order to avoid collisions (conflicts over common radio resources), yet

maintaining the swarm cohesion (i.e., avoiding unnecessary spread in the occu-

pancy of the time-frequency plane). This procedure is applied to the dynamic

resource allocation in the frequency domain and in the time-frequency domain,

where the primary users in a cognitive radio system are modeled as statistically

independent homogeneous continuous-time Markov processes.

A rigorous mathematical analysis of the proposed algorithm is also derived. First,

we study the stability and the cohesiveness of the swarm in case of local inter-

actions among the nodes, providing closed form expressions for the upper and

lower bounds of the swarm size as a function of the network connectivity. Then,

using stochastic approximation arguments, we derive the convergence properties

of the swarming algorithm in the presence of random disturbances introduced by

realistic channels, i.e., link failures, quantization, noise and estimation errors.

Spectrum sensing is a critical prerequisite in envisioned applications of wire-

less cognitive radio networks which promise to resolve the perceived bandwidth

scarcity versus under-utilization dilemma. Creating an interference map of the

operational region plays an instrumental role in enabling spatial frequency reuse

and allowing for dynamic spectrum allocation in a hierarchical access model com-

prising primary and secondary users. For such purpose, a distributed technique

for cooperative spectrum estimation in cognitive radio systems is proposed based

on a basis expansion model of the power spectral density map in frequency. The

proposed method, based on diffusion adaptation algorithms, estimates and learns

the interference profile through local cooperation and without the need for a

central processor. Convergence and mean square analysis of the diffusion filter

applied to the distributed cooperative sensing problem is also derived.

4

Finally, it is proposed a dynamic resource allocation technique combining a dis-

tributed diffusion algorithm, for implementing cooperative sensing, with a swarm-

ing technique, for allocating resources in a parsimonious way (i.e., avoiding un-

necessary spread in the frequency domain), yet avoiding collisions. In particular,

the procedure is applied to the dynamic resource allocation problem in the fre-

quency domain. Numerical results show the improvement that results in the

resource allocation performance due to the cooperative estimation of the spec-

trum. Furthermore, it is shown how the proposed technique endows the resulting

bio-inspired network with powerful learning and adaptation capabilities.

5

Acknowledgements

Mi ritengo fortunato di aver conosciuto tutte le persone che mi hanno accom-

pagnato in questa splendida avventura del dottorato di ricerca.

In primis, vorrei ringraziare il mio mentore, il Professor Sergio Barbarossa,

per la sua guida ed il suo continuo insegnamento durante questi anni. Mi sento

estremamente onorato della fiducia e della stima che ha sempre riposto in me.

Non e mai esistito un giorno in cui non mi abbia dedicato tempo ed attenzione,

indipendentemente da quanto occupato fosse. La sua straordinaria motivazione,

il suo intuito e la sua profonda conoscenza tecnica sono state per me fonte di

grande ispirazione. Spero che le mie capacita siano state il piu possibile plasmate

da tali eccezionali qualita.

I wish to thank the supervisor of my work at UCLA, Professor Ali H. Sayed.

He welcomed me in his research group and treated me like another member of the

group with no distinction, making my stay at Los Angeles a very pleasant and

rewarding experience. And of course, I would like to thank all the guys of the

Adaptive System Laboratory at UCLA: Zaid, Alex, Xiaochuan, Jianshu, Shine,

Shang Kee, Victor and Jae-Woo. I will never forget the friendship with which

they honored me during my stay.

I miei ringraziamenti vanno anche i miei colleghi Marco, Stefania, Alessandro

e Pasquale con cui ho condiviso le esperienze ed il lavoro di tutti i giorni in

un clima sereno e piacevole. Desidero anche ringraziare in modo particolare

7

Gesualdo Scutari per avermi sempre incoraggiato ed entusiasmato nel perseguire

l’eccellenza del mio lavoro fin dall’inizio del mio dottorato.

Di certo in questa lista di ringraziamenti non posso scordare i miei amici

di una vita con i quali da sempre condivido tutte le gioie e i dolori. Marco,

Roberto, Marco, Christian, Luca, Daniele, Valerio, Simone, Marco, Massimiliano.

Ognuno di voi ha contribuito ogni giorno a rendermi un uomo migliore e mi ritengo

estremamente fortunato di aver potuto sempre contare su amici veri come voi.

Un ringraziamento speciale lo dedico al maestro Giampiero che mi ha ritenuto

degno di poter prendere il suo posto. Mi ha sempre insegnato tutto quello che

sapeva al meglio che poteva e, di questo, gliene saro sempre grato. Vorrei anche

includere Francesco, Ciro e Matteo, per ringraziarli dell’impegno che ogni giorno

dedicano a cio che e parte fondamentale della mia vita.

Il cammino della vita d’improvviso mi ha fatto incontrare te, Anna. La tua

bellezza, intelligenza, forza e dolcezza hanno completato la mia vita come non

credevo possibile. Grazie alla nostra splendida sinergia, niente mi spaventa se

sono con te ad affrontarlo. Oggi, cosı come domani, ti ringraziero sempre di

camminare con me sul sentiero della vita e di essere la mia felicita.

Infine, ringrazio mio padre Bernardo, mia madre Rosalba e mio fratello An-

drea, le persone da cui ho imparato di piu. Il loro incondizionato amore, inseg-

namento e supporto sono stati ineguagliabili durante tutta la mia vita.

8

Contents

Abstract 3

1 Introduction and Overview 17

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Network Design Inspired by Biological Models . . . . . . . . . . . . 26

1.2.1 Modeling Approaches . . . . . . . . . . . . . . . . . . . . . 27

1.2.2 Classification and Categorization . . . . . . . . . . . . . . . 28

1.3 Bio-Inspired Networking . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.1 Swarm Intelligence . . . . . . . . . . . . . . . . . . . . . . . 29

1.3.2 Firefly Synchronization . . . . . . . . . . . . . . . . . . . . 32

1.3.3 Activator-Inhibitor Systems . . . . . . . . . . . . . . . . . . 33

1.3.4 Artificial Immune System . . . . . . . . . . . . . . . . . . . 35

1.3.5 Epidemic Spreading . . . . . . . . . . . . . . . . . . . . . . 37

1.3.6 Nano-scale and Molecular Communication . . . . . . . . . . 39

1.4 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.5 Research Contributions . . . . . . . . . . . . . . . . . . . . . . . . 45

2 Mathematical Background 49

2.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.1.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 54

9

CONTENTS

2.2 Distributed Optimization . . . . . . . . . . . . . . . . . . . . . . . 60

2.2.1 Unconstrained Optimization . . . . . . . . . . . . . . . . . . 60

2.2.2 Constrained Optimization . . . . . . . . . . . . . . . . . . . 62

2.2.3 Convex Constrained Optimization Problems . . . . . . . . . 63

2.3 Stochastic Approximation . . . . . . . . . . . . . . . . . . . . . . . 66

2.3.1 Robbins-Monro procedure . . . . . . . . . . . . . . . . . . . 67

2.3.2 Kiefer-Wolfowitz procedure . . . . . . . . . . . . . . . . . . 71

2.4 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.4.1 Directed Graphs: The Basic Mathematical Tool to De-

scribe Interactions . . . . . . . . . . . . . . . . . . . . . . . 74

2.4.2 Algebraic Graph Theory . . . . . . . . . . . . . . . . . . . . 78

3 Distributed Resource Allocation Based on Swarming Mecha-

nisms 83

3.1 Introduction on Cognitive Radio and Dynamic Radio Access . . . 84

3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.3 Continuous-Time Distributed Optimization . . . . . . . . . . . . . 94

3.4 Stability and Cohesion Analysis . . . . . . . . . . . . . . . . . . . . 97

3.4.1 Profiles with Bounded Gradient . . . . . . . . . . . . . . . . 98

3.4.2 Quadratic Profile . . . . . . . . . . . . . . . . . . . . . . . . 107

3.5 Swarming in the Frequency Domain . . . . . . . . . . . . . . . . . 110

3.5.1 Local Stability Analysis . . . . . . . . . . . . . . . . . . . . 111

3.5.2 Discrete-Time Implementation . . . . . . . . . . . . . . . . 112

3.5.3 Fast Swarming Algorithms . . . . . . . . . . . . . . . . . . 114

3.5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . 116

3.6 Swarming in the Time-Frequency Domain . . . . . . . . . . . . . . 126

3.6.1 Swarming in a Static Interference Environment . . . . . . . 127

3.6.2 Swarming in the Presence of Markovian Interference . . . . 130

3.7 Discrete-Time Distributed Optimization . . . . . . . . . . . . . . . 132

3.7.1 Projected Swarming Algorithms . . . . . . . . . . . . . . . 135

10

CONTENTS

3.8 The Effect of Noise and Realistic Channels . . . . . . . . . . . . . 137

3.8.1 Random Link Failures . . . . . . . . . . . . . . . . . . . . . 139

3.8.2 Dithered Quantization . . . . . . . . . . . . . . . . . . . . . 139

3.8.3 Stochastic Convergence . . . . . . . . . . . . . . . . . . . . 140

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4 Distributed Cooperative Spectrum Sensing Based on Diffusion

Adaptation 171

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.2 Diffusion Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.3 Sparse Diffusion Adaptation . . . . . . . . . . . . . . . . . . . . . . 176

4.3.1 Sparse ATC Diffusion . . . . . . . . . . . . . . . . . . . . . 177

4.3.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . 179

4.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 183

4.4 Basis Expansion Model of the Spectrum . . . . . . . . . . . . . . . 187

4.5 ATC Diffusion for Adaptive Spectrum Estimation . . . . . . . . . . 190

4.5.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . 191

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5 Swarming for Dynamic Radio Access Based on Diffusion Adap-

tation 199

5.1 Swarm Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

5.2 Diffusion Adaptation for Cooperative Spectrum Sensing . . . . . . 202

5.3 Adaptive Swarming . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

6 Concluding Remarks 215

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Bibliography 221

11

List of Figures

2.1 Fixed points with three different types of stability. The fixed point

on the left is stable. The fixed point in the center is marginally

stable. The fixed point on the right is unstable. . . . . . . . . . . . 57

2.2 Examples of graphs: (a) Strongly connected graph. (b) Quasi

strongly connected graph with one root strongly connected com-

ponent and two strongly connected components. (c) WC graph

containing a two-tree forest. . . . . . . . . . . . . . . . . . . . . . . 78

3.1 Magnitude of the coupling function g(·) in (3.4) with linear attrac-

tion (3.7) and unbounded repulsion (3.8), using the values cA = 1

and cR = 2. The distance between the red points and zero is the

equilibrium distance between the swarm agents. . . . . . . . . . . . 96

3.2 Magnitude of the coupling function g(·) in (3.4) with linear attrac-

tion (3.7) and bounded repulsion (3.10), using the values cA = 1,

cR = 10 and cG = 2. The distance between the red points and

zero is the equilibrium distance between the swarm agents. . . . . 97

3.3 Upper and lower bounds of the potential function time derivative. 104

3.4 Swarm size parameter versus the node covering radius. . . . . . . . 116

3.5 Swarm size parameter versus the attraction parameter cA. . . . . . 117

3.6 Network topology and allocation example. . . . . . . . . . . . . . . 118

3.7 Interference profile and allocation example. . . . . . . . . . . . . . 119

13

LIST OF FIGURES

3.8 Frequency reuse parameter versus covering radius. . . . . . . . . . 120

3.9 Normalized system potential function vs. time index, for different

coverage radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.10 Normalized system potential function vs. time index, for different

descent directions of the algorithm : gradient descent (dashed) and

Newton approximation (solid). . . . . . . . . . . . . . . . . . . . . 122

3.11 Dynamic resource allocation by swarming: Reaction time to PU’s

activations, for basic swarming. . . . . . . . . . . . . . . . . . . . . 123

3.12 Dynamic resource allocation by swarming: Reaction time to PU’s

activations, for adaptive scaling. . . . . . . . . . . . . . . . . . . . 124

3.13 Average number of iterations to obtain convergence versus number

of nodes, for different degrees of network connectivity. . . . . . . . 125

3.14 Example of 2D allocation, considering a quadratic profile. . . . . . 127

3.15 Swarm size versus the magnitude of the quadratic profile Aσ. . . . 128

3.16 Swarm size versus the magnitude of the quadratic profile Aσ. . . . 129

3.17 Example of time-frequency allocation. . . . . . . . . . . . . . . . . 130

3.18 Example of time-frequency allocation with Markovian interference. 132

3.19 Secondary network. The square nodes denote primary users and

the circle nodes denote secondary users. . . . . . . . . . . . . . . . 158

3.20 Examples of resource allocation by swarming. . . . . . . . . . . . . 159

3.21 Average interference perceived by the swarm vs. time index, for

different probabilities of correct packet reception. . . . . . . . . . . 160

3.22 Average interference perceived by the swarm at convergence, ver-

sus the probability to establish a communication link, for different

values of the swarm attraction parameter cA. . . . . . . . . . . . . 161

3.23 Average convergence time versus number of nodes, for different

number of bits used for quantization. . . . . . . . . . . . . . . . . . 162

3.24 Average convergence time versus number of nodes, for different

degrees of network connectivity. . . . . . . . . . . . . . . . . . . . . 163

14

LIST OF FIGURES

3.25 Average interference perceived by the swarm vs. time index, for

different algorithms and probabilities of correct packet reception. . 164

3.26 Average interference perceived by the swarm at convergence, ver-

sus the slope parameter of the linear scaling functions, for different

probabilities of correct packet reception and different values of the

attraction parameter cA. . . . . . . . . . . . . . . . . . . . . . . . . 165

3.27 Example of resource allocation by swarming. . . . . . . . . . . . . 167

4.1 Transient network MSD for the non-cooperative approaches LMS

[190], ZA-LMS [166], RZA-LMS [166], and the diffusion techniques

ATC [145], ZA-ATC (eq.(4.3)-(4.9)), RZA-ATC (eq.(4.3)-(4.11)). . 184

4.2 Differential MSD versus sparsity parameter ρ for ZA-ATC Diffu-

sion LMS, for different degrees of system sparsity. . . . . . . . . . . 185

4.3 Differential MSD versus sparsity parameter ρ for RZA-ATC Dif-

fusion LMS, for different degrees of system sparsity. . . . . . . . . 186

4.4 Example of basis expansion using Gaussian pulses. The dotted

curves represent the Gaussian basis functions, whereas the contin-

uous curve denotes the behavior of a generic interference profile

described by 6 Gaussian pulses. . . . . . . . . . . . . . . . . . . . . 188

5.1 Secondary network. The square nodes denote primary users and

the dot nodes denote secondary users. . . . . . . . . . . . . . . . . 206

5.2 Comparison of the result of spectrum estimation through coop-

erative diffusion adaptation and without cooperation among the

users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.3 Steady-state MSD versus node index. . . . . . . . . . . . . . . . . . 208

5.4 Average interference perceived by the swarm at convergence, for

the non cooperative estimation case and for adaptive diffusion. . . 209

5.5 Different resource assignments in dynamic environment. . . . . . . 210

5.6 ATC diffusion learning curve, in terms of MSD. . . . . . . . . . . . 211

5.7 Average perceived interference versus iteration index. . . . . . . . . 212

15

Chapter 1

Introduction and Overview

1.1 Motivation

The last few decades have witnessed striking developments in communication

and networking technologies, yielding many information network architectures.

These next generation information networks are envisioned to be characterized

by an invisible and ubiquitous halo of information and communication services,

which should be easily accessible by users in a transparent, location-independent,

and seamless fashion. Therefore, the result will be a pervasive and, in fact, liv-

ing network. This ubiquitous networking space will include, in addition to the

traditional Internet-connected devices, networked entities that are in much closer

interaction with us such as wearable networks [6], in-body molecular communi-

cation networks [7], unattended ground, air, and underwater sensor networks [8],

self-organizing sensor and actor networks [9, 10] and locally intelligent and self-

cognitive devices exploiting the communication resources with the help of cogni-

tive capabilities, e.g., cognitive radio networks [92]. Clearly, this vision implies

that almost every object will be able to effectively and collaboratively communi-

cate, thus becoming, to some extent, a node of the future pervasive global net-

work. The evolution in communication and networking technologies brings many

17

1.1 Motivation

potential advantages to our daily lives. At the same time, the complexity of the

existing and envisioned networked information systems is rapidly going beyond

what conventional networking paradigms can do. Thus, self-organization tech-

niques are demanded to overcome current technical limitations [11]. In fact, there

exist many common significant challenges that need to be addressed for practi-

cal realization of existing and next generation networking architectures, such as

increased complexity with large scale networks, their dynamic nature, resource

constraints, heterogeneous architectures, absence or impracticality of centralized

control and infrastructure, need for survivability, and unattended resolution of

potential failures.

Most of the existing and next generation communication systems, handled ac-

cording to the conventional networking paradigms, do not totally accommodate

the scale, heterogeneity and complexity of such scenarios. Then, novel paradigms

are needed for designing, engineering and managing these communication sys-

tems. While the challenges outlined above such as scalability, heterogeneity and

complexity are somehow new byproducts of the evolution in the communication

technologies in the last few decades, they have been successfully dealt with by Na-

ture for quite some time. Unlike the evolution in the communication technologies

that have brought these challenges, the evolution in Nature has shown how bio-

logical systems can actually handle many of these challenges with an elegance and

efficiency still far beyond current techniques. In fact, when we look carefully into

Nature, it is clearly observed that the dynamics of many biological systems and

laws governing them are based on a surprisingly small number of simple generic

rules, which yield collaborative yet effective patterns for resource management

and task allocation, social differentiation, synchronization (or de-synchronization)

without the need for any externally controlling entity. For example, by means

of these capabilities, billions of blood cells that constitute the immune system

can protect the organism from the pathogens without the central control of the

brain [20]. Similarly, the biological homeostasis autonomously maintains the op-

eration of vital functions of an entire organism without any need for a central

18

1.1 Motivation

biological controller [21]. Furthermore, the task allocation process in the insect

colonies is collaboratively decided and performed such that the overall task is op-

timized with a global intelligence comprised of simple individual responses [22].

These examples and, in general, as a result of millions of years of evolution, bi-

ological systems and processes have intrinsic appealing characteristics. Among

others, they are:

• adaptive to the varying environmental circumstances,

• robust and resilient to the failures caused by internal or external factors,

• able to achieve complex behaviors on the basis of a usually limited set of

basic rules,

• able to learn and evolve itself when new conditions are applied,

• effective in managing constrained resources with an apparently global in-

telligence larger than the superposition of individuals,

• able to self-organize in a fully distributed fashion, collaboratively achieving

an efficient equilibrium,

• survivable despite harsh environmental conditions due to its inherent and

sufficient redundancy.

These characteristics lead to different levels of inspiration from biological sys-

tems towards the derivation of different algorithm designs for efficient, robust

and resilient communication networks. Therefore, in order to keep pace with

the evolution in-networking technologies, many researchers are currently engaged

in developing innovative design paradigms inspired by biology in order to ad-

dress the networking challenges of existing and envisioned information systems.

The common rational behind this effort is to capture the governing dynamics

and understand the fundamentals of biological systems in order to devise new

methodologies and tools for designing and managing communication systems

19

1.1 Motivation

and information networks that are inherently adaptive to dynamic environments,

heterogeneous, scalable, self-organizing, and evolvable. Besides bio-inspired net-

working solutions, communication on the nano-scale is being investigated with

two important but conceptually different goals. On the one hand, bio-inspired

nano machinery is investigated in order to build machines on the nano level using

communication and actuation capabilities derived from biological counterparts.

More specifically, the most promising communication mechanism between nano-

machines forming nano-scale networks is currently envisioned to be molecular

communication, i.e., coding and transfer of information in terms of molecules,

which is also mainly inspired by the cellular signaling networks observed in living

organisms. On the other hand, such nano-machines can also be used in the main

field of molecular biology to study biological systems. Clearly, there exist many

challenges for the realization of the existing and the envisioned next generation

network architectures. At the same time, we would like to stress that several

biological approaches may be used as a solution of these networking paradigms.

In this section, we review some of the most challenging issues for networking and

highlight the analogies with their counterparts and corresponding solutions that

already biological systems offer.

Large Scale Networking

One of the main challenges is related to the increasing size exhibited by the

networking systems, which connect a huge numbers of users and devices in a sin-

gle, omni-comprehensive, preferably always-on network. The size of this network,

in terms of both number of constituent nodes and running services, is expected to

exceed by several orders of magnitude that of the current Internet. For example,

Wireless Sensor Networks (WSNs), having a broad range of current and future

applications, are generally envisioned to be composed of a large number, e.g., in

numbers ranging between few hundreds to several hundred thousands, of sensor

nodes [12]. The first direct consequence of such large scales is the huge amount

20

1.1 Motivation

of traffic load to be incurred over the network, which could easily exceed the

network capacity, hampering the communication reliability due to packet losses

by both collisions in the local wireless channel as well as congestion along the

network path [14]. Similarly, it becomes more important to find the optimal

routes, in order to keep the communication overhead at acceptable levels during

the dissemination of a large amount of information over a large scale network. As

the network scale expands, the number of possible paths, and hence, the search

space for the optimal route in terms of a preset criteria, also drastically enlarges.

Hence, networking mechanisms must be scalable and adaptive to variations in the

network size. Actually, there exist many biological systems that inspire the design

of effective communication solutions for large scale networks. For example, Ant

Colony Optimization (ACO) techniques [23] provide efficient routing mechanisms

for large-scale mobile ad hoc networks and information dissemination over large

scales can be handled using epidemic spreading [24], which is the transmission

mechanism of viruses.

Dynamic Nature

Unlike the early communication systems composed of a transmitter/receiver

pair and a communication channel, which are all static, the existing and future

networking architectures are highly dynamic in terms of node behaviors, traffic

and bandwidth demand patterns, channel and network conditions. According to

the mobility of the nodes, network dimensions, and radio ranges, communica-

tion links may frequently be established and become obsolete in mobile ad hoc

networks [16]. Furthermore, due to mobility of the nodes, and environmental

variations as a result of movement, the channel conditions and hence link qual-

ities may be highly dynamic. Dynamic spectrum access and its management in

cognitive radio networks is an important case where the dynamic nature of the

user behaviors poses significant challenges on the network design [92]. The objec-

tive of cognitive radio networks itself is to exploit the dynamic usage of spectrum

21

1.1 Motivation

resources in order to maximize the overall spectrum utilization.

To this end, biological systems are known to be capable of adapting them-

selves to varying circumstances. For example, Artificial Immune System (AIS),

inspired by the principles and processes of the mammalian immune system [25],

efficiently detects variations in the dynamic environment or deviations from the

expected system patterns. Similarly, activator-inhibitor systems and the analysis

of reaction-diffusion mechanisms in biological systems [26] also capture dynam-

ics of highly interacting systems through differential equations. These specific

models can be exploited to develop communication techniques that can adapt to

varying environmental conditions.

Resource Constraints

As the communication technologies evolve, network demands also increase in

terms of available services, service quality and lifetime. For example, the cur-

rent Internet can no longer respond to every demand as its capacity is almost

exceeded by the total traffic created, which lays a basis for the development of

next generation Internet [17]. At the same time, with the increased demand

from wireless networking, fixed spectrum assignment-based traditional wireless

communications has become insufficient in accommodating a wide range of radio

communication requests. Consequently, cognitive radio networks with dynamic

spectrum management and access has been proposed and is currently being de-

signed in order to improve utilization of spectrum resources [92]. More specif-

ically, for the networks composed of nodes that are inherently constrained in

terms of energy and communication resources, e.g., WSNs [12], Mobile Ad Hoc

Networks (MANETs) [16], nano-scale and molecular communication networks [7],

these limitations directly bound their performance and mandate for intelligent re-

source allocation mechanisms. The biological systems yet again help researchers

by providing potential solutions that address the trade-off between the high de-

mand and limited supply of resources. For example, in the foraging process [23],

22

1.1 Motivation

ants use their individual limited resources towards optimizing the global behavior

of colonies in order to find food source in a cost-effective way. The behavior of ant

colonies in the foraging process may inspire many resource-efficient networking

techniques. Furthermore, cellular signaling networks represent and capture the

dynamics of interactions contributing to the main function of a living cell. Hence,

they might also be exploited in order to obtain efficient communication techniques

for resource constrained nano-scale and molecular communication networks.

Need for Infrastructure-less and Autonomous Operation

The significant increase in network dimensions, both spatially and in the

number of nodes, makes centralized control of the network very unpractical.

On the other hand, some networks are by definition free from infrastructure

such as wireless ad hoc networks [16], Delay Tolerant Networks (DTNs) [17],

WSNs [12], and some have a heterogeneous, mostly distributed and non-unified

system architecture such as cognitive radio networks [92], wireless mesh networks

and WiMAX [13]. These networking environments mandate for distributed com-

munication and networking algorithms which can effectively work without any

help from a centralized unit. At the same time, communication networks are

subject to failure either by device malfunction, e.g., nodes in a certain area may

run out of battery in sensor networks, or misuse of their capacity, e.g., overloading

the network may cause heavy congestion blocking the connections. In most cases,

networks are expected to continue their operation without any interruption due to

these potential failures. Considering the dynamic nature, lack of infrastructure,

and impracticality of centralized communication control, it is clear that networks

must be capable of self-organization and self-healing in order to be able to resume

their operation. Hence, the existing and next generation information networks

must have the capabilities of self-organization, self-evolution and survivability.

In order to address all these needs, networks could exploit some intelligent algo-

rithms and processes that were largely observed in biological systems. In fact,

23

1.1 Motivation

inherent features of many biological systems stand as promising solutions for these

challenges. For example, an epidemic spreading mechanism could be modified for

efficient information dissemination in highly partitioned networks and for oppor-

tunistic routing in delay tolerant networking environments [24]. Ant colonies, and

in general insect colonies, which perform global tasks without the control of any

centralized entity, could also inspire the design of communication techniques for

infrastructureless networking environments [76]. Furthermore, synchronization

principles of fireflies [27] could be applied to the design of time synchronization

protocols as well as communication protocols requiring precise time synchroniza-

tion. Activator-inhibitor systems may be exploited for distributed control of

sensing periods and duty cycle of target tracking sensor networks [28]. The au-

tonomous behavior of artificial immune systems may be a good model for the

design of effective algorithms for unattended and autonomous communication in

sensor networks. Thus, the lack of infrastructure and autonomous communica-

tion requirements in various networking environments could be addressed through

careful exploration of self organization capabilities of biological systems.

Heterogeneous Architectures

The other critical aspect of many of the existing and envisioned communica-

tion networks is linked to their heterogeneity and its resultant extremely complex

global behavior, emerging from the diverse range of network elements and large

number of possible interactions among them. Next generation communication

systems are generally envisioned to be composed of a vast class of communi-

cating devices differing in their communication/storage/processing capabilities,

ranging from Radio Frequency Identification (RFID) devices and simple sensors

to mobile vehicles equipped with broadband wireless access devices. Similarly,

cognitive radio networks involve the design of new communication techniques

to realize the co-existence of different wireless systems communicating on over-

lapping spectrum bands with an ultimate objective of maximizing the spectrum

24

1.1 Motivation

utilization. Wireless mesh networks and WiMAX are also expected to be com-

posed of heterogeneous communication devices and algorithms [13]. Sensor and

Actor Networks (SANETs) architecturally incorporate both heterogeneous low-

end sensor nodes and highly capable actor nodes [11]; and Vehicular Ad Hoc

Networks (VANETs) [29] exhibit significant levels of heterogeneity in terms of

wireless communication technologies in use and mobility patterns of ad hoc vehi-

cles. Such heterogeneity and asymmetry in terms of capabilities, communication

devices and techniques need to be understood, modeled and effectively managed,

in order to allow the realization of heterogeneous novel communication networks.

Different levels of heterogeneity are also observed in biological systems. For

example, in many biological organisms, despite external disturbances, a stable

internal state is maintained through collaborative effort of heterogeneous set of

subsystems and mechanisms, e.g., nervous system, endocrine system, immune

system. This functionality is called homeostasis, and the collective homeostatic

behavior [78] can be applied towards designing communication techniques for

networks with heterogeneous architectures. On the other hand, insect colonies

are composed of individuals with different capabilities and abilities to respond to

a certain environmental stimuli. Despite this inherent heterogeneity, colonies can

globally optimize the task allocation via their collective intelligence [23]. Similar

approaches can be adopted to address task assignment and selection in SANETs,

for spectrum sharing in heterogeneous cognitive radio networks [76], as well as

multi-path routing in overlay networks [30].

Communication on the Micro Level

With the advances in micro- and nano-technologies, electro-mechanical de-

vices have been downscaled to micro and nano-levels. Consequently, there ex-

ist many micro (MEMS) and nano-electro-mechanical systems (NEMS) with a

large spectrum of applications. Clearly, capabilities for communication and net-

working at micro and even at nano-scales become imperative in order to enable

25

1.2 Network Design Inspired by Biological Models

micro and nano-devices to cooperate and, hence, collaboratively realize certain

common complex task that cannot be handled individually. In this regard, nano-

networks could be defined as a network composed of nano-scale machines, i.e.,

nano-machines, cooperatively communicating with each other and sharing infor-

mation in order to fulfill a common objective [79]. The dimensions of nano-

machines render conventional communication technologies, such as electromag-

netic or acoustic waves, inapplicable at these scales due to antenna size and

channel limitations. Furthermore, the communication medium and channel char-

acteristics also show important deviations from the traditional cases due to the

rules of physics governing these scales. The main idea of nano-machines and nano-

scale communications and networks have also been motivated and inspired by the

biological systems and processes. Hence, it is conceivable that the solutions for

the challenges in communication and networking at micro and nano-scales could

also be developed through inspiration from the existing biological structures and

communication mechanisms. In fact, many biological entities in organisms have

similar structures with nano-machines. For example, every living cell has the

capability of sensing the environment, receiving external signals, performing cer-

tain tasks at nano-scales. More importantly, based on transmission and reception

of molecules, cells in a biological organism may establish cellular signaling net-

works [80], through which they can communicate in order to realize more complex

and vital tasks, e.g., immune system responses. Therefore, the inspiration from

cellular signaling networks, and hence, molecular communication, provide impor-

tant research directions and promising design approaches for communication and

networking solutions at micro- and nano-scales.


The main intention of this first chapter is to introduce and to overview the

emerging area of bio-inspired networking. Therefore, the scope of this section is

first to introduce the general approach to bio-inspired networking by discussing

26


the identification of biological structures and techniques relevant to communica-

tion networks, modeling the systems and system properties, and finally deriving

optimized technical solutions. Secondly, we try to classify the field of biologi-

cally inspired approaches to networking. Bio-inspired algorithms can effectively

be used for optimization problems, exploration and mapping, and pattern recog-

nition. Based on several examples, we will see that bio-inspired approaches have

outstanding capabilities and potential applications that motivate our interest.

1.2.1 Modeling Approaches

Before introducing the specific biological models that have been exploited to-

wards the development and realization of bio-inspired networking solutions, we

briefly illustrate the general modeling approach. First modeling approaches date

back to the early 1970ies [31, 32], since, that time, quite a number of technical

solutions mimicking biological counterparts have been developed and published.

Typical bio-networking architectures showing the complete modeling approach

are described in [33, 34]. This bio-networking architecture can be seen as a cat-

alyzer or promoter for many other investigations in the last decade. Looking at

many papers that have been published in recent years, the main effort was focused

on presenting technical solutions with some similarities to biological counterparts

without really investigating the key advantages or objectives of the biological sys-

tems. Obviously, many methods and techniques are really bio-inspired as they

follow principles that have been studied in Nature and that promise positive ef-

fects if applied to technical systems. Three steps can be identified that are always

necessary for developing bio-inspired methods that have a remarkable impact in

the domain under investigation:

1. Identification of analogies - similar structures and methods,

2. Understanding - detailed modeling of realistic biological behavior,

3. Engineering - model simplification and tuning for technical applications.

27


First, analogies between biological and technical systems, such as computing and

networking systems, must be identified. It is especially necessary that all the

biological principles are understood properly, which is often not yet the case in

biology. Secondly, models must be created for the biological behavior. These

models will later be used to develop the technical solution. The translation

from biological models to the model describing bio-inspired technical systems is

a pure engineering step. Finally, the model must be simplified and tuned for the

technical application. As a remark, it should be mentioned that biologists already

started looking at bio-inspired systems to learn more about the behavioral pattern

in nature [36]. Thus, the loop closes from technical applications to biological

systems.

1.2.2 Classification and Categorization

Basically, the following application domains of bio-inspired solutions to prob-

lems related to computing and communications can be distinguished:

• Bio-inspired computing represents a class of algorithms focusing on efficient

computing, e.g. for optimization processes and pattern recognition.

• Bio-inspired systems constitute a class of system architectures for massively

distributed and collaborative systems, e.g. for distributed sensing and ex-

ploration.

• Bio-inspired networking is a class of strategies for efficient and scalable

networking under uncertain conditions, e.g. for autonomic organization in

massively distributed systems.

Looking from biological principles, several application domains in networking

can be distinguished. Besides these specific algorithms that are mimicking biolog-

ical mechanisms and behavior, the general organization of biological systems, i.e.

the structure of bodies down to organs and cells, can be used as an inspiration

to develop scalable and self-organizing technical systems.

28

1.3 Bio-Inspired Networking


In this section, we introduce some examples of bio-inspired networking devel-

oped in last years. Of course, the following list is not meant to be comprehensive

and to completely represent all approaches in the domain of bio-inspired network-

ing. However, we selected a number of techniques and methods for more detailed

presentation that clearly show advantages in fields of communication networks.

In the discussion, we try to highlight the necessary modeling of biological phe-

nomena or principles and their application in networking.

1.3.1 Swarm Intelligence

Coordination principles studied in the fields of swarm intelligence [22] and es-

pecially those related to social insects give insights into principles of distributed

coordination in Nature. In many cases, direct communication among individ-

ual insects is exploited, e.g., in the case of dancing bees [39] or social foraging

swarms [118]. However, especially the stigmergic communication via changes in

the environment is as fascinating as helpful to coordinate massively distributed

systems. For example, Ma and Krings studied the chemosensory communication

systems in many of the moth, ant and beetle populations [38]. The difference

between the wireless network of an insect population and an engineered wireless

sensor network is that insects encode messages with semiochemicals (also known

as infochemicals) rather than with radio frequencies. Application examples of the

bees’ dance range from routing to intruder detection [39]. Another typical ex-

ample is the communication between ants for collaborative foraging. Ant Colony

Optimization (ACO) is perhaps the best analyzed branch of swarm intelligence

based algorithms. In general, swarm intelligence is based on the observation of

the collective behavior of decentralized and self-organized systems such as ant

colonies, flocks of fish, or swarms of bees or birds [22]. Such systems are typically

made up of a population of simple agents interacting locally with one another

and with their environment. In most cases, swarm intelligence based algorithms

29


are inspired by the behavior of foraging ants [22]. Ants are able to solve complex

tasks by simple local means. There is only indirect interaction between individ-

uals through modifications of the environment, e.g. pheromone trails are used

for efficient foraging. ACO is based on the principles of the foraging process of

ants. Ants perform a random search (random walk) for food. The way back to

the nest is marked with a pheromone trail. If successful, the ants return to the

nest (following their own trail). While returning, an extensive pheromone trail

is produced pointing towards the food source. Thus, further ants will follow and

reinforce the trail on the shortest path towards the food. The ants therefore

communicate based on environmental changes (pheromone trail), i.e. they use

stigmergic communication techniques for communication and collaboration. The

complete ACO algorithm is described in [40]. The most important aspect in this

algorithm is the transition probability pkij for an ant k to move from i to j. This

probability represents the routing information for the exploring process

pkij =

(

ταij(t)× ηβij

)

/(

∑

l∈Jkiταil (t)η

βil

)

, if j ∈ Jki ;

0, otherwise.(1.1)

Each move depends on the following parameters:

• Jki is the “tabu” list of not yet visited nodes, i.e., by exploiting Jk

i , an ant

k can avoid visiting a node i more than once.

• ηij is the visibility of j when standing at i, i.e., the inverse of the distance.

• τij is the pheromone level of edge (i; j), i.e., the learned desirability of

choosing node j and currently at node i.

• α and β are adjustable parameters that control the relative weight of the

trail intensity τij and the visibility ηij , respectively.

After completing a tour, each ant k lays a quantity of pheromone ∆τkij on each

edge (i; j) according to the following rule, where T k(t) is the tour done by ant

30


k at iteration t, Lk(t) is its length, and Q is a parameter (which only weakly

influences the final result)

∆τkij =

Q/Lk(t), if (i; j) ∈ T k(t);

0, otherwise.(1.2)

Dynamics in the environment are explicitly considered by the ant foraging scheme.

The pheromone slowly evaporates. Thus, if foraging ants are no longer successful,

the pheromone trail will dissolve and the ants continue with their search process.

Additionally, randomness is also a strong factor during successful foraging. A

number of ants will continue the random search for food. This adaptive behavior

leads to an optimal search and exploration strategy. This effect is provided by

the pheromone update rule, where ∆τij =∑m

k=1∆τkij. The decay is implemented

in form of a coefficient ρ with 0 < ρ < 1.

τij(t)← (1− ρ)× τij(t) + ∆τij(t) (1.3)

According to [40], the total number of ants m is an important parameter of the

algorithm. Too many ants would quickly reinforce suboptimal tracks and lead

to early convergence to bad solutions, whereas too few ants would not produce

enough decaying pheromone to achieve the desired cooperative behavior. Thus,

the decay rate needs to be carefully controlled.

The main applications in networking that are based on the main concepts of

ACO are briefly reported in the following.

1. Routing : The best known examples of ACO in networking are the AntNet

[41] and AntHocNet [40] routing protocols. Both protocols follow the con-

cepts of ant routing. In particular, so called agents are used to concurrently

explore the network and exchange collected information in the same way

as ants explore the environment. The communication among the agents is

indirect, following the stigmergy approach, and mediated by the network

itself.

31


2. Task Allocation : Based on the same concepts, integrated task allocation

and routing in SANETs has been investigated [81]. The proposed architec-

ture is completely based on probabilistic decisions. During the lifetime of

the SANET, all nodes maintain and adapt a probability P (i) to execute a

task out i of a given set. Reinforcement strategies are exploited to optimize

the overall system behavior. It needs to be mentioned that the integrated

task allocation and routing approach represents a typical cross-layer solu-

tion. Application layer and network layer are both responsible for operating

the entire SANET.

3. Search in Peer-2-Peer networks : Search in Peer-2-Peer (P2P) net-

works is usually provided by centralized or decentralized lookup tables.

However, the effort to find data in unstructured decentralized P2P net-

works can easily become the dominating factor. Also in this case, it is

expected the use of ant-based approaches [42].

1.3.2 Firefly Synchronization

Precise synchronization in massively distributed systems is a complex issue

and hard to achieve. Recently, new models for clock synchronization have been

proposed based on the synchronization principles of fireflies. In this context, early

biological experiments have been conducted by Richmond, who also discovered

the underlying mathematical synchronization model [27]. Basically, the firefly

synchronization is based on pulse-coupled oscillators [43]. The simple model for

synchronous firing of biological oscillators consists of a population of identical

integrate-and-fire oscillators. A local variable xi is integrated from zero to one

and the oscillator fires when xi = 1. Then, the xi jumps back to zero.

dxidt

= S0 − γxi. (1.4)

Multiple oscillators are assumed to interact in form of simple pulse coupling:

when a given oscillator fires, it pulls the others up by a fixed amount ǫ, bringing

32


them toward the firing threshold.

xi(t) = 1 ∀j 6= i : xj(t+) = min(1;xj(t) + ǫ) (1.5)

As a result, for almost all initial conditions the population evolves to a state in

which all the oscillators are firing synchronously. The presented concept of self-

organized clock synchronization has been successfully applied to synchronization

in ad hoc networks [58, 59]. Using a linearly incrementing phase function φi, the

local pulse of a node is controlled: when φi reaches a threshold φth, the local

oscillator fires. For a period of T , this can be described as follows:

dφidt

=φthT. (1.6)

When coupling identical oscillators, the phase can be controlled according to

Equation (1.5). Additional effort is needed to compensate the transmission de-

lays in ad hoc and sensor networks. This can be done by selecting appropriate

values for ǫ. In particular, the phase shift is dynamically updated according to

the estimated transmission delay. The general application of this clock synchro-

nization technique for wireless networks is discussed in [44].

1.3.3 Activator-Inhibitor Systems

The basis for exploiting the characteristics of activator-inhibitor systems in

technical systems is the analysis of reaction-diffusion mechanisms. In the 1950ies,

the chemical basis of morphogenesis has been analyzed [45]. The underlying

reaction and diffusion in a ring of cells has been successfully described in form

of differential equations. Assuming that for concentrations of X and Y chemical

reactions are tending to increase X at the rate f(X;Y ) and Y at the rate of

g(X;Y ), the changes of X and Y due to diffusion also take into account the

behavior of the entire system, i.e. all the neighboring N cells. Thus, the rate

of such chemical reactions can be described by the 2N differential equations [45]

(where r = 1, . . . , N , µ is the diffusion constant for X and ν is the diffusion

33


constant for Y ):

dXr

dt= f(Xr;Yr) + µ(Xr+1 −X2r +Xr−1) (1.7)

dYrdt

= g(Xr;Yr) + ν(Xr+1 −X2r +Xr−1) (1.8)

For general application (independent of the shape of the generated pattern or the

structure of interacting systems), this set of differential equations can be written

as (with F and G being nonlinear functions for (chemical) reactions, Du and Dv

describe the diffusion rates of activator and inhibitor, and ∇2 is the Laplacian

operator):

du

dt= f(u; v)−Du∇

2u (1.9)

dv

dt= g(u; v) −Dv∇

2v (1.10)

Reaction-diffusion pattern formation is used to support high-level tasks in smart

sensor networks. In particular, on-off patterns in large-scale deployments for for-

est fire scenarios have been investigated. As a key result, different shapes have

been detected such as stripes, spots, and ring patterns, that can be exploited for

high-level activities such as navigating robots to the source of the fire. Further

experiments and considerations on reaction-diffusion based pattern generation

in sensor networks are described in [46]. Again, reaction-diffusion based con-

trol mechanisms have been investigated. Similarly, cooperative control can be

achieved based on a reaction-diffusion equation for surveillance system [49]. As

can be seen from the mentioned approaches, sensor coordination is one of the

primary application fields for employing activator-inhibitor mechanisms. In the

following, two further solutions are depicted that coordinate sensing activities in

WSNs to achieve improved energy performance, i.e. to maximize the network

lifetime [48]. In [47], pattern formation models are used to coordinate the on-off

cycles of sensor nodes. In particular, sensors are allowed to control their sen-

sory and their radio transceiver while, at the same time, the network needs to

be able to transmit sensor data over a multi-hop network to one or more data

34


sinks. Each sensor stores its own activator and inhibitor values and broadcasts

them every τ seconds. Using the received data, the neighboring nodes re-evaluate

the reaction-diffusion equations. Sensors with a activator value exceeding some

(given) threshold become active by turning on their sensing circuitry. As shown

in [47], the performance of the system achieves astonishingly good results. Sim-

ilarly, the distributed control of processing periods is investigated in [28]. Using

the programming system Rule-based Sensor Network (RSN), a sensor network is

configured for target tracking. In this example, the duty cycle is controlled by a

promoter/inhibitor system that takes into account the efficiency of the local ob-

servations and the results from neighboring nodes. By exploiting the information

transmitted towards a sink node, each node can estimate the need for further

local measurements and adequately update the local sampling period.

1.3.4 Artificial Immune System

The term Artificial Immune System (AIS) refers to adaptive systems inspired

by theoretical and experimental immunology with the goal of problem solving [50].

The primary goal of an AIS, which is inspired by the principles and processes

of the mammalian immune system [25], is to efficiently detect changes in the

environment or deviations from the normal system behavior in complex problems

domains. The role of the mammalian immune system can be summarized as

follows: It protects the body from infections by continuously scanning for invading

pathogens, e.g. exogenous (non-self) proteins. AIS based algorithms typically

exploit the immune system’s characteristics of self-learning and memorization.

The immune system is, in its simplest form, a cascade of detection and adaptation,

culminating in a system that is remarkably effective. In Nature, two immune

responses were identified. The primary one is to launch a response to invading

pathogens leading to an unspecific response (using Leucoytes). In contrast, the

secondary immune response remembers past encounters, i.e. it represents the

immunologic memory. It allows a faster response the second time by showing a

35


very specific response (using B-cells and T-cells). An AIS basically consists of

three parts, which have to be worked out in the immune engineering process [50]:

• Representations of the system components, i.e. the mapping of technical

components to antigens and antibodies.

• Affinity measures, i.e. mechanisms to evaluate interactions (e.g., stimula-

tion pattern and fitness functions) and the matching of antigens and anti-

bodies.

• Adaptation procedures to incorporate the system’s dynamics, i.e. genetic

selection.

A first AIS has been developed by Kephart [51], and early approaches showing

the successful application of such AISs in computer and communication systems

have been presented in [25]. Meanwhile, a number of frameworks are available.

Focusing on the design phase of an AIS, de Castro and Timmis [50] proposed an

immune engineering framework. A similar conceptual frameworks for Artificial

Immune Systems for generic application in networking has been presented in [52].

Again, three steps for designing the framework have been emphasized: represen-

tation, selection of appropriate affinity measures, and development of immune

algorithms. In this framework, Markov chains are used to describe the system’s

dynamics. Data analysis and anomaly detection represent typical application

domains [50]. The complete scope of AISs is widespread. Sample applications

have been developed for fault and anomaly detection, data mining (e.g., ma-

chine learning, pattern recognition), agent based systems, control, and robotics.

Pioneering work by Timmis and coworkers conceptually analyzed the AIS and ap-

plied it to several problem domains [20,52]. An application of an immune system

based distributed node and rate selection in sensor networks has been proposed

in [15]. Sensor networks and their capabilities, in particular their transmission

rate, are modeled as antigens and antibodies. The distributed node and rate

selection (DNRS) algorithm for event monitoring and reporting is achieved by B-

36


cell stimulation, i.e. appropriate node selection. This stimulation depends on the

following influences: (1) the affinity between the sensor node (B-cell) and event

source (pathogen), (2) the affinity between the sensor node and its uncorrelated

neighbor nodes (stimulating B-cells), and (3) the affinity between the sensor node

and its correlated neighbor nodes (suppressing B-cells).

1.3.5 Epidemic Spreading

Epidemic Spreading is frequently used as an analogy to understand the in-

formation dissemination in wireless ad hoc networks. Information dissemination

in this context can refer to the distribution of information particles (as usually

provided by ad hoc routing techniques) [24] or to the spread of viruses in the

Internet [53] or on mobile devices [55]. Biological models of virus transmission

provide means for assessing such emerging threats and to understand epidemics

as a general purpose communication mechanism. A number of mathematical

models of different networks have been investigated, which lie at various points

on a broad conceptual spectrum. At one end there are network models reflect-

ing strong spatial effects, with nodes at fixed positions in two dimensions, each

connected to a small number of neighbors. At the other end there are scale-free

networks, which are essentially unconstrained by physical proximity, and in which

the number of contacts per node are widely spread. The main difference is in the

epidemic spread. In scale-free networks, epidemics can persist at arbitrarily low

levels, whereas in simple two-dimensional models a minimum level of virulence

is needed to prevent them from dying out quickly [55]. The system model for

epidemic communication relies on a population, i.e. a number of nodes that rep-

resent the network. Information entities are exchanged among the nodes using

a diffusion algorithm. All transmissions are usually assumed to be atomic, i.e.

there will be no split during diffusion. Then, all the nodes can be distinguished

into two groups: susceptible nodes, S(t) describes this set at a certain time t, and

infective nodes I(t) [56]. The diffusion algorithm is then a process that converts

37


susceptible nodes into infective nodes with a rate α = βxN I(t), where β is the

probability of information transmission, i.e. the infection probability, x describes

the number of contacts among susceptible nodes, and N is the total number of

nodes. The infection rate can then be described as:

dI

dt= α× S(t) =

βx

NI(t)× S(t) (1.11)

A measure for the connectedness of the nodes is termed eigenvector centrality.

Let us consider a graph model of the network topology and denote by A the

corresponding adjacency matrix. The eigenvector centrality of a node i is defined

being proportional to the sum of the eigenvector centralities of i’s neighbors,

where e represents the vector of nodes’ centrality scores. Otherwise stated, e is

the eigenvector of A relative to the eigenvalue λ:

ei = Ae/λ. (1.12)

Depending on the particular application scenario, the healing rate, i.e. the non-

negative rate of converting infective nodes, also needs to be considered in this

equation. There is a wide application range for epidemic communication in com-

puter networks. Primarily, the focus is on routing in mobile ad hoc networks with

growing interest in opportunistic routing, in which messages are passed between

devices that come into physical proximity, with the goal of eventually reaching

a specified recipient. For example, the understanding of the spread of epidemics

in highly partitioned mobile networks has been studied in [57]. The main appli-

cation field in this work was the use of epidemic communication in DTNs. As a

conclusion, the paper outlines the possibility to roughly measure the importance

of a node to the process of epidemic spreading by the node’s eigenvector central-

ity. Regions, as defined by the steepest-ascent rule, are clusters of the network in

which spreading is expected to be relatively rapid and predictable. Furthermore,

nodes whose links connect distinct regions play an important role in the (less

rapid, and less predictable) spreading from one region to another. The charac-

teristics of epidemic information dissemination have been carefully modeled to

38


investigate the inherent characteristics [58]. For example, the buffer management

plays and important role and a stepwise probabilistic buffering has been proposed

as a solution [59]. Detailed models have been built to study the performance im-

pact of epidemic spreading [60]. Whereas Markov models lead to quite accurate

performance predictions, the numerical solution becomes impractical if the num-

ber of nodes is large. In [60], an unified framework based on ordinary differential

equations is presented that provides appropriate scaling as the number of nodes

increases. This approach allows the derivation of closed-form formulas for the

performance metrics while obtaining matching results compared to the Markov

models. The interesting result is that the network topology plays an important

role whether epidemics can be applied for improved robustness and efficiency. In

particular, the scale-free property must be ensured in order to overcome possible

problems with transmissions that quickly die out. A slightly different problem

(and solution) has been addressed in [61]. The targeted question is that the prob-

lem of determining the right information collection infrastructure can be viewed

as a variation of the network design problem including additional constraints such

as energy efficiency and redundancy. As the general problem is NP-hard, the au-

thors propose a heuristic based on the mammalian circulatory system, which

results in a better solution to the design problem than the state-of-the-art alter-

natives. The resulting system for wireless sensor networks is quite similar to the

epidemics approach even though only the communication within an organism is

used as an analogy. Besides efficient routing solutions, the application to network

security is maybe the most important aspect of epidemic models.

1.3.6 Nano-scale and Molecular Communication

Incredible improvements in the field of nano-technologies have enabled nano-

scale machines that promise new solutions for several applications in biomedical,

industry and military fields. Some of these applications might exploit the po-

tential advantages of communication and hence cooperative behavior of these

39


nano-scale machines to achieve a common and challenging objective that exceeds

the capabilities of a single device. At this point, the term nano-networks is de-

fined as a set of nano-scale devices, i.e., nano-machines, communicating with each

other and sharing information to realize a common objective. Nano-networks

allow nano-machines to cooperatively communicate and share any kind of in-

formation such as odor, flavor, light, or any chemical state in order to achieve

specific tasks required by wide range of applications including biomedical en-

gineering, nuclear, biological, and chemical defense technologies, environmental

monitoring. Despite the similarity between communication and network func-

tional requirements of traditional and nano-scale networks, nano-networks bring

a set of unique challenges. In general, nano-machines can be categorized into

two types: one type mimics the existing electro-mechanical machines and the

other type mimics nature-made nano-machines, e.g., molecular motors and re-

ceptors. In both types, the dimensions of nano-machines render conventional

communication technologies such as electromagnetic wave, acoustic, inapplicable

at these scales due to antenna size and channel limitations. In addition, the avail-

able memory and processing capabilities are extremely limited, which makes the

use of complex communication algorithms and protocols impractical in the nano

regime. Furthermore, the communication medium and the channel characteris-

tics also show important deviations from the traditional cases due to the rules of

physics governing these scales. For example, due to size and capabilities of nano-

machines, traditional wireless communication with radio waves cannot be used to

communicate between nano-machines, which may constitute of just several moles

of atoms or molecules and scale on the orders of a few nanometers. Hence, these

unique challenges need to be addressed in order to effectively realize the nano-

scale communication and nano-networks in many applications from nano-scale

body area networks to nano-scale molecular computers. The motivation behind

nano-machines and nano-scale communications and networks have also originated

and been inspired by the biological systems and processes. In fact, nano-networks

are significant and novel artifacts of bio-inspiration in terms of both their archi-

40


tectural elements, e.g., nano-machines, and their communication mechanism, i.e.,

molecular communication. Indeed, many biological entities in organisms have

similar structures with nano-machines, i.e., cells, and similar interaction mech-

anism and vital processes, cellular signaling [80], with nano-networks. Within

cells of living organisms, nano-machines called molecular motors, such as dynein,

myosin [62], realize intracellular communication through chemical energy trans-

formation. Similarly, within a tissue, cells communicate with each other through

the release over the surface and the diffusion of certain soluble molecules, which

are received by a specific receptor molecule on another cell. Apparently, cellu-

lar signaling networks are the fundamental source of inspiration for the design

of nano-networks. Therefore, the communication and networking problems in

nano-networks may also be inspired by the similar biological processes. The

main communication mechanism of cellular signaling is based on transmission

and reception of certain type of molecules, i.e., molecular communication, which

is, indeed, the most promising and explored communication mechanism for nano-

networks. In Nature, molecular communication between biological entities takes

place according to the ligand receptor binding mechanism. Ligand molecules are

emitted by one biological phenomenon; then, the emitted ligand molecules dif-

fuse in the environment and bind the receptors of another biological phenomenon.

This binding enables the biological phenomenon to receive the bound molecules

by means of the diffusion on cell membrane. The received ligand molecules allow

the biological phenomenon to understand the biological information. For exam-

ple, in a biological endocrine system, gland cells emit hormones to intercellular

environment; then, hormone molecules diffuse and are received by correspond-

ing cells. According to the type of emitted hormone, the corresponding cells

convert the hormone molecule to biologically meaningful information. This nat-

ural mechanism provides the molecular communication for almost all biological

phenomena. Following the main principles of this mechanism, a number of stud-

ies have been performed on the design of nano-scale communication. Molecular

communication and some design approaches are introduced [64], and its funda-

41


mental research challenges are first manifested in [65]. Different mechanisms are

proposed for molecular communication including a molecular motor communi-

cation system [66], intercellular calcium signaling networks [63], an autonomous

molecular propagation system to transport information molecules using DNA hy-

bridization and bio-molecular linear motors. An information theoretical analysis

of a single molecular communication channel is performed in [7]. An adaptive er-

ror compensation mechanism is devised for improving molecular communication

channel capacity in [67]. In [68], molecular multiple access, relay and broadcast

channels are modeled and analyzed in terms of capacity limits and the effects of

molecular kinetics and environment on the communication performance are inves-

tigated. Based on the use of vesicles embedded with channel forming proteins,

a communication interface mechanism is introduced for molecular communica-

tion in [69, 70]. In addition, a wide range of application domains of molecular

communication based nano-networks are introduced from nano-robotics to future

health-care systems [71]. Clearly, inspired by biological systems, molecular com-

munication, which enables nano-machines to communicate with each other using

molecules as information carrier, stands as the most promising communication

paradigm for nano-networks. While some research efforts and initial set of results

exist in the literature, many open research issues remain to be addressed for the

realization of nano-networks. Among these, first is the thorough exploration of

biological systems, communications and processes, in order to identify different

efficient and practical communication techniques to be exploited by innovative

nano-network designs. The clear set of challenges for networked communication

in nano-scale environments must be precisely determined for these different po-

tential bio-inspired solution avenues. Applicability of the traditional definitions,

performance metrics and well-known basic techniques, e.g., Time Division Mul-

tiple Access (TDMA), random access, minimum cost routing, retransmission,

error control, congestion, must be studied. Furthermore, potential problems for

the fundamental functionalities of nano-networks, such as modulation, channel

coding, medium access control, routing, congestion control, reliability, must be

42

1.4 Outline of Dissertation

investigated without losing the sight of the bio-inspired perspective in order to

develop efficient, practical and reliable nano-network communication techniques

through inspiration from the existing biological structures and communication

mechanisms.


The outline of each chapter is as follows.

Chapter 1, the present chapter, gives the motivation, outline and contributions

of this dissertation. A brief survey on bio-inspired networking is also presented

to enlighten the potential benefits offered by bio-inspired solutions.

Chapter 2 introduces several important theories on which many results of this

dissertation are based. In particular, section 2.1 recalls some basic results from

dynamical systems theory, focusing on the stability analysis of linear and nonlin-

ear systems. In section 2.2 we analyze the main descent methods used to optimize

a cost function in a distributed manner, reporting classical convergence results.

Then, in section 2.3, we recall stochastic approximation methods aimed at finding

zeros or maxima of an objective function, whose value measurable at each time

instant is corrupted by additive noise, providing basic convergence results. The

last section is devoted to graph theory, reviewing the notation and basic results

that will be largely used in this dissertation.

Chapter 3 proposes a bio-inspired radio access mechanism for cognitive networks

mimicking the behavior of a flock of birds swarming in search for food in a cohe-

sive fashion without colliding with each other. The equivalence between swarming

and radio resource allocation is established by modeling the interference distribu-

tion in the resource domain, e.g. frequency and time, as the spatial distribution

of food, while the position of the single bird represents the radio resource chosen

by each radio node. The solution is given as the distributed minimization of a

43


functional, borrowed from social foraging swarming models, containing the aver-

age interference plus repulsion and attraction terms that help to avoid conflicts

and maintain cohesiveness, respectively. A stability and cohesion analysis is de-

rived under different assumptions on the attraction/repulsion terms, showing the

effect played by the swarm parameters and connectivity on the final swarm size.

Several examples illustrate how the proposed method can be applied to dynamic

resource allocation on the frequency domain and the time-frequency domain, pro-

viding an intrinsic capability of the system to provide spatial reuse of frequency,

through a purely decentralized mechanism. In the last part of the chapter, we

also consider the swarming algorithm in the presence of channel imperfections,

such as link failures, estimation errors, and quantization noise. Thus, we derive

the almost sure convergence of the swarming procedure to an equilibrium con-

figuration dependent on the mean graph of the network, even in the presence of

such random disturbances.

Chapter 4 develops an adaptive algorithm for spectrum estimation in cogni-

tive radio networks based on diffusion adaptation algorithms. In particular, we

address this task through a parsimonious basis expansion model of the PSD in

frequency. This model reduces the sensing task to estimating a common vector of

unknown parameters. The resulting estimator relies on diffusion adaptation al-

gorithms, where the cognitive radios exchange information locally only with their

one-hop neighbors, eliminating the need for a fusion center. First, we describe

the basic diffusion algorithm, then we introduce novel regularized diffusion LMS

techniques for distributed estimation over adaptive networks, which are able to

exploit sparsity in the underlying system model. Convergence and mean square

analysis of the sparse adaptive diffusion filter show under what conditions we

have dominance of the proposed method with respect to its unregularized coun-

terpart in terms of steady-state performance. Simulation results also confirm the

potential benefits of the proposed filter under the sparsity assumption on the

true coefficient vector. Exploiting these estimation schemes, we illustrate the

44

1.5 Research Contributions

proposed distributed spectrum estimation technique based on diffusion adapta-

tion. We first introduce a basis expansion model, which is useful to model the

PU’s transmission, allowing distributed cooperative sensing. Then, we propose a

normalized version of the Adapt then Combine (ATC) diffusion algorithm, which

enables the network to learn and track the time-varying interference profile. Con-

vergence and mean-square performance analysis of the proposed normalized ATC

diffusion filter, applied to the spectrum estimation problem, is also derived.

Chapter 5 studies the learning abilities of adaptive networks in the context of

cognitive radio networks and investigates how well they assist in allocating power

and communications resources in the frequency domain. The allocation mech-

anism is based on a social foraging swarm model that lets every node allocate

its resources (power/bits) in the frequency regions where the interference is at a

minimum while avoiding collisions with other nodes. We employ adaptive diffu-

sion techniques to estimate the interference profile in a cooperative manner and

to guide the motion of the swarm individuals in the resource domain. The result-

ing bio-inspired network cooperatively estimates the interference profile in the

resource domain of a cognitive network and allocates resources through purely

decentralized mechanisms. Finally, the resulting procedure is applied to the dy-

namic resource allocation problem in the frequency domain. Numerical results

show the improvement that results in the resource allocation performance due

to the cooperative estimation of the spectrum. Furthermore, it is shown how

the proposed technique endows the resulting bio-inspired network with powerful

learning and adaptation capabilities.

Chapter 6 concludes the dissertation summarizing the main obtained results.


The main contribution of this dissertation is the development of a bio-inspired

resource allocation technique for dynamic radio access in cognitive radio systems.

45


Details of the research contributions in each chapter are as follows.

Chapter 3

The main result in this chapter is regarding the bio-inspired formulation of

the resource allocation problem published (or to be published) in two journal

papers and three conference papers:

• P. Di Lorenzo and S. Barbarossa, “A bio-inspired swarming algorithm for

decentralized access in cognitive radio,” IEEE Transactions on Signal Pro-

cessing, vol. 59, no. 12, December 2011.

• P. Di Lorenzo and S. Barbarossa, “Decentralized resource assignment in

cognitive networks based on swarming mechanisms over random graphs,”

accepted for publication in IEEE Transactions on Signal Processing.

• P. Di Lorenzo and S. Barbarossa, “Distributed resource allocation in cog-

nitive radio systems based on social foraging swarms,” in Proceedings of

the 11th IEEE International Workshop on Signal Processing Advances in

Wireless Communications, Marrakech, pp. 1-5, June 2010. (IEEE best

student paper award)

• P. Di Lorenzo and S. Barbarossa, “Bio-inspired swarming models for de-

centralized radio access incorporating random links and quantized commu-

nications,” Proceedings of the 36th International Conference on Acoustics,

Speech and Signal Processing , pp. 5780-5783, Prague, May 2011. (Invited

to the special session: Bio-inspired Information Processing and Networks).

• P. Di Lorenzo, S. Barbarossa, and Ali H. Sayed “A bio-inspired fast swarm-

ing algorithm for dynamic radio access,” in Proceedings of the 17th Interna-

tional Conference on Digital Signal Processing , Corfu, Greece, July 2011,

pp. 1-6. (Invited to the special session: Signal Processing for Cognitive

Radio).

46


Chapters 4 and 5

The main results in these chapters regard the development of a totally adap-

tive bio-inspired network, based on swarming and diffusion adaptation mecha-

nisms, that senses, learns and reacts to changes in the environment. The research

contributions are published (or will be published) in two journal papers and two

conference papers:

• P. Di Lorenzo, Ali H. Sayed, and S. Barbarossa “Bio-inspired dynamic radio

access based on swarming mechanisms over adaptive networks,” submitted

to IEEE Transactions on Signal Processing.

• P. Di Lorenzo, Ali H. Sayed, and S. Barbarossa “Sparse distributed estima-

tion over adaptive networks,” submitted to IEEE Transactions on Signal

Processing.

• P. Di Lorenzo, S. Barbarossa, and Ali H. Sayed “Bio-inspired swarming for

dynamic radio access based on diffusion adaptation,” in Proceedings of the

19-th European Signal Processing Conference, Barcelona, Spain, August-

September 2011, pp. 402-406. (EURASIP best student paper award)

• P. Di Lorenzo, S. Barbarossa, and Ali H. Sayed “Sparse diffusion LMS for

distributed adaptive estimation,” in Proc. of the International Conference

on Acoustics, Speech and Signal Processing, Kyoto, Japan, March 2012.

Other contributions not presented in this dissertation

During the author’s Ph.D. period, optimal beamforming techniques were de-

veloped for ambiguity suppression in squinted synthetic aperture radar systems.

The results are published (or will be published) in one journal paper and one

conference paper:

• P. Di Lorenzo, S. Barbarossa, and Leonardo Borgarelli “Optimal beamform-

ing for range-doppler ambiguity minimization in squinted SAR,” accepted

47


for publication in IEEE Transactions on Aerospace and Electronic Systems.

• P. Di Lorenzo and S. Barbarossa “Optimal beamforming for range-doppler

ambiguity suppression in squinted SAR systems,” in Proceedings of the 4th

IEEE International Workshop on Computational Advances in Multi-Sensor

Adaptive Processing, San Juan, Puerto Rico, pp. 169-172, December 2011.

(IEEE best student paper award)

Some work was also done in the area of distributed resource allocation in

femtocell networks systems with results published (or to be published) in one

journal paper and one conference paper:

• P. Di Lorenzo, S. Barbarossa and Marco Omilipo, “Distributed sum-rate

maximization over finite rate coordination links affected by random fail-

ures,” submitted to IEEE Transactions on Signal Processing.

• P. Di Lorenzo, Marco Omilipo and S. Barbarossa “Distributed stochas-

tic pricing for sum-rate maximization in femtocell networks with random

graph and quantized communications,” in Proceedings of the 4th IEEE In-

ternational Workshop on Computational Advances in Multi-Sensor Adap-

tive Processing, San Juan, Puerto Rico, pp. 165-168, December 2011.

Furthermore, some work was done in the field of signal processing for wireless

sensor networks with results published (or to be published) in one journal paper

and one conference paper:

• Sergio Barbarossa, Stefania Sardellitti and Paolo Di Lorenzo, “Distributed

estimation and detection with applications in Wireless Sensor Networks,”

submitted to Elsevier Signal Processing e-reference.

• P. Di Lorenzo and S. Barbarossa, “Wireless Sensor Networks With Dis-

tributed Decision Capabilities Based On Self-Synchronization Of Relax-

ation Oscillators,” in Proc. IEEE International Wireless Communications

and Mobile Computing Conference 2008, Creta, August 2008, pp. 45-49.

48

Chapter 2

Mathematical Background

In this chapter we recall some basic mathematical tools that will be largely

used throughout this work. First, in section 2.1, we consider dynamical systems,

both in continuous and discrete time, giving the basic analytic tools to prove

stability in the linear and nonlinear case. Section 2.2 introduces iterative methods

for the solution of nonlinear problems, focusing on descent methods and reporting

several convergence results in the unconstrained and constrained case. Then, in

section 2.3, we recall stochastic approximation methods aimed at finding zeros or

maxima of an objective function, whose value measurable at each time instant is

corrupted by additive noise, providing basic convergence results. The last section

is devoted to graph theory, reviewing the notation and basic results that will be

largely used in this work.

2.1 Dynamical Systems

In this section we introduce and illustrate some important concepts of dy-

namical system theory. A dynamical system consists of a set of variables that

describe its state and a law that describes the evolution of the state variables

with time, i.e., how the state of the system in the next moment of time depends

on the input and its state in the previous moment of time. The evolution law is

49


given by a system of ordinary differential equations. Mathematically, a dynam-

ical system is specified by a state vector x ∈ Rn, (a list of numbers which may

change as time progresses) and a function f : Rn → Rn, which describes how the

system evolves over time. There are two kinds of dynamical systems: discrete

time and continuous time. For a discrete time dynamical system, we denote time

by k, and the system is specified by the equations

x[k + 1] = f(x[k]), x[0] = x0. (2.1)

It thus follows that x[k] = fk(x0), where fk denotes a k-fold application of f

to x0. For a continuous time dynamical system, we denote time by t, and the

following equations specify the system:

x(t) = f(x(t)), x(0) = x0. (2.2)

where x is a time-dependent vector variable denoting the current state of the

system, x(t) is its derivative with respect to time t, f is a scalar function that

determines the evolution of the system, x0 ∈ Rn is an initial condition. In this

case, Rn is called phase space or state space to stress the fact that each state of the

system corresponds to a certain point in Rn. When all parameters are constant,

the dynamical system is called autonomous. When at least one of the parameters

is time-dependent, the system is non-autonomous. To solve (2.2) means to find

a function x(t) whose initial value is x[0] = x0 and whose derivative is f(x(t))

at each moment t ≥ 0. Finding explicit solutions is often impossible even for

such simple systems, so quantitative analysis is carried out mostly via numerical

simulations. The simplest procedure to solve numerically, known as first-order

Euler method, replaces (2.2) by the discretized system

x(t+ h)− x(t)

h= f(x(t)), (2.3)

where t = 0, h, 2h, 3h... is the discrete time and h is a small time step. Knowing

the current state x(t), we can find the next state point via

x(t+ h) = x(t) + hf (x(t)). (2.4)

50


Iterating this difference equation starting with x[0] = x0, we can approximate the

analytical solution of (2.2). The approximation has a noticeable error of order h,

so scientific software packages, such as MATLAB, use more sophisticated high-

precision numerical methods. In many cases, however, we do not need exact

solutions, but rather qualitative understanding of the behavior of (2.2) and how

it depends on parameters and the initial state x0. For example, we might be

interested in the number of equilibrium (rest) points the system could have,

whether the equilibria are stable, their attraction domains, etc.

In the following we present some classical results from dynamical system the-

ory that will be used throughout the paper. First, we consider linear systems,

extending then our attention to nonlinear systems. The following sections deal

with fixed points and their stability. In particular, we consider two methods for

assessing stability: linearization and Lyapunov functions.

2.1.1 Linear Systems

Previously, we introduced discrete and continuous time dynamical systems.

The function f : Rn → Rn might be quite simple or terribly complicated. In

this section we study dynamical systems in which the function f is particularly

nice: we assume f is linear. Then, considering the general vector case, we have

f(x) = Ax + b, where A ∈ Rn × R

n is an n × n matrix and b ∈ Rn is a fixed

n-dimensional vector.

Discrete-Time

In this section we consider linear discrete time dynamical systems of the form:

x[k + 1] = Ax[k] + b, x[0] = x0. (2.5)

In our analysis, we begin by dropping the ”+b” term in (2.5) and concentrating

on the system x[k + 1] = Ax[k]. Now, it is very easy to check that

x[k] = Akx0. (2.6)

51


To go further, we consider the case in which the matrix A diagonalizes. We

assume thatA has n linearly independent eigenvectors u1, . . . ,un with associated

eigenvalues λ1, . . . , λn . Let Λ be the diagonal matrix with diagonal entries

λ1, . . . , λn , and let U ∈ Rn × R

n be the n× n matrix whose i-th column is ui .

Thus, we may write A = UΛU−1. Notice that

Ak = (UΛU−1)(UΛU−1) . . . (UΛU−1). (2.7)

Now, since matrix multiplication is associative and the terms U−1U evaluate to

I, we have

Ak = UΛkU−1. (2.8)

Now, Λ is a diagonal matrix whose diagonal entries are the A’s eigenvalues:

λ1, . . . , λn. Raising a diagonal matrix to a power is an easy task, achieving

Λk = diag[λk1 , . . . , λkn]. Thus, to understand the behavior of the system, we

need to study the eigenvalues of the iteration matrix A. In particular, if all the

eigenvalues of A have absolute value less than 1, thenAk tends to the zero matrix

as k → ∞. Thus x[k] = Akx0 → 0. On the other hand, if some eigenvalue of

A has absolute value greater than 1, entries in Ak are diverging to ∞. Let’s

examine how this affects the values x[k]. We are assuming that the eigenvectors

u1, . . . ,un are linearly independent. Any family of n linearly independent vectors

in Rn forms a basis, hence every vector (x0 in particular) can be written uniquely

as a linear combination of the ui’s. Thus we may write

x0 = c1u1 + . . .+ cnun, (2.9)

where the ci’s are scalar numbers. Now, since each ui is an eigenvector of A, we

have Aui = λiui, thus Akui = λki ui. The iterative process in (2.6) can then be

written in terms of eigenvalues and eigenvectors of the matrix A as

x[k] = Akx0 = c1λk1u1 + . . .+ cnλ

knun. (2.10)

Then, if all the eigenvalues of A all have absolute value less than 1, then x[k]→ 0

as k → ∞. If some eigenvalue has absolute value bigger than 1, then typically

52


|x[k]| → ∞. For some very special x0 (those with ci = 0 if |λi| > 1), x[k] does

not explode. Finally, if some eigenvalues have absolute value equal to 1, and

the rest have absolute value less than 1, then typically x[k] neither explodes nor

vanishes but dances about at a modest distance from 0. The situation for the

more general case x[k+1] = Ax[k]+b is quite similar. As before, it is possible to

determine the iterative pattern of the dynamical system in (2.5). In particular,

provided that the matrix I−A is invertible (A does not have an eigenvalue equal

to one), we have

x[k] = Akx0 + (I −Ak)(I −A)−1b. (2.11)

Then, if the absolute values of A’s eigenvalues are all less than 1 (hence I −A

is invertible), then Ak tends to the zero matrix, hence x[k] → x = (I −A)−1b.

Alternatively, if some eigenvalues have absolute value bigger than 1, then Ak

blows up, and for most x0 we have |x[k]| → ∞. (There are exceptional x0 ’s, of

course. For example, if 1 is not an eigenvalue of A and if x0 = x = (I −A)−1b,

then x[k] = x for all k. Finally, if some eigenvalues have absolute value equal

to 1 and the other eigenvalues have absolute value less than 1, we see a range of

behaviors and the system might stay near x, or it might blow up.

Continuous-Time

Now we turn to continuous-time multidimensional linear systems, that is,

systems of the form

x(t) = Ax(t) + b, x(0) = x0. (2.12)

As before, we begin our study of multidimensional continuous-time linear systems

with a simplification, namely, that b = 0. The system becomes x(t) = Ax(t).

This is a system of n differential equations in n variables. These equations are

difficult to solve because each is dependent on the other. However, it is possible

to show that the system x(t) = Ax(t) has nearly as simple a solution, which is

given by

x(t) = exp(At)x0. (2.13)

53


Then, to understand the behavior of the system, we need to study how exp(At)

behaves. Assuming that A diagonalizes as A = UΛU−1, after some algebra it is

possible to show that

exp(At) = U exp(Λt)U−1, (2.14)

where

exp(Λt) =

eλ1t 0 . . . 0

0 eλ2t . . . 0...

.... . .

...

0 0 . . . eλnt

. (2.15)

Also in this case, the behavior of the system depends on the eigenvalues of the

matrix A. It follows, therefore, that the individual components of x (i.e., x1(t)

through xn(t)) are linear combinations of eλ1t, . . . , eλnt.

Considering also the possibility to have complex eigenvalues, we report the

following general principle: we have that x(t) → 0 as t → ∞ if the real parts

of all of A’s eigenvalues are negative; if some eigenvalue has positive real part,

then typically |x(t)| → ∞; and if Reλ ≤ 0 for all λ, but Reλ ≥ 0 for some λ,

then x(t) neither settles down to any specific value nor does it blow up. For the

more general case x(t) = Ax(t) + b, we have similar results. In particular, we

have that x(t)→ x = −A−1b as t→∞ if the real parts of all of A’s eigenvalues

are negative; if some eigenvalue has positive real part, then typically |x(t)| → ∞;

and if Reλ ≤ 0 for all λ, but Reλ ≥ 0 for some λ, then x(t) stays near x but

does not approach it, or does it blow up.

2.1.2 Nonlinear Systems

The general forms for dynamical systems are

x(t) = f(x(t)) for continuous-time, and

x[k + 1] = f(x[k]) for discrete-time.

We have previously considered the case when f(·) is linear. In that case, we can

answer nearly any question we might consider. We can work out exact formulas

54


for the behavior of x(t) (or x[k]) and deduce from them the long-term behavior

of the system. There are two main behaviors: (1) the system gravitates toward

a fixed point, or (2) the system blows up. There are some marginal behaviors

as well. Now we begin our study of more general systems in which f(·) can

be virtually any function. However, in this section, we do make the following

assumption: f(·) is differentiable with continuous derivative. Nonlinear systems

are more complicated; we seek qualitative descriptions in place of exact formulas.

Indeed, typically, it is impossible to find exact formulas for x. Further, the range

of behaviors available to nonlinear systems is much greater than that for linear

systems. Because it can be terribly difficult to find exact solutions to nonlinear

systems, our goal reduces in determining the long-term behavior of the system.

This is often feasible even when finding an exact solution is not. In the following

we focus on the notion of a fixed point (sometimes called an equilibrium point)

of a dynamical system. We also discuss how to to determine if they are stable or

unstable. Often, understanding the fixed points of a dynamical system can tell

us much about the global behavior of the system.

Fixed Points

The vector x is the state of the dynamical system, and the function f(·) tells

us how the system moves. In special circumstances, however, the system does

not move. The system can be stuck (we will say fixed) in a special state; we

call these states fixed points of the dynamical system. Thus a fixed point of a

dynamical system is a state vector x with the property that if the system is ever

in the state x, it will remain in that state for all time. Finding fixed points of

dynamical systems does not require us to find exact formulas for x[k] or x(t).

All we have to do is solve a system of equations. Of course, solving systems of

equations can be difficult, but it is at least comforting to know that this is the

only issue involved. The equations we solve depend on f(·) and whether the

system is in discrete or continuous time. In discrete time we solve x = f(x), and

55


in continuous time we solve f(x) = 0.

Not all fixed points are the same. Let us now describe three types of fixed

points a system may possess:

• A fixed point x is called stable provided the following is true: For all starting

values x0 near x, the system not only stays near x but also x(t) → x as

t→∞ [or x[k]→ x as k →∞ in discrete time];

• A fixed point x is called marginally stable or neutral provided the following:

For all starting values x0 near x, the system stays near x but does not

converge to x;

• A fixed point x is called unstable if it is neither stable nor marginally stable.

In other words, there are starting values x0 very near x so that the system

moves far away from x.

Figure 2.1 illustrates each of these possibilities. The fixed point on the left of

the figure is stable; all trajectories which begin near x remain near, and converge

to, x. The fixed point in the center of the figure is marginally stable (neutral).

Trajectories which begin near x stay nearby but never converge to x. Finally,

the fixed point on the right of the figure is unstable. There are trajectories which

start near x and move far away from x.

A question is still open. Once we have found fixed points, how can we tell

whether they are stable or unstable? The next sections will provide some tools

for making this determination. First, we consider linearization methods, where

we approximate our system near its fixed points by linear functions. Then, we

introduce also the general method of using Lyapunov functions to determine the

stability of dynamical systems.

Linearization

The purpose of this section is to provide a method to tell whether a fixed

point x of a dynamical system [either discrete or continuous time] is stable or

56


Figure 2.1: Fixed points with three different types of stability. The fixed point

on the left is stable. The fixed point in the center is marginally stable. The fixed

point on the right is unstable.

unstable. If the function f(·) is linear, i.e., of the form f(x) = Ax + b, the

answer is relatively easy: We check the eigenvalues of A (either their absolute

values or their real parts, depending on the nature of time). Then, the method

of linearization is based on the approximation of f(·) near its fixed point x by

using a linear function. Since f(·) : Rn → Rn, the approximation we seek is of

the form

f(x) = J(x)(x− x) + f(x), (2.16)

where J(x) : Rn×Rn is an n×n matrix which gives the best approximation. In

particular, if we write

f(x) = J(x)(x− x) + f(x) + error(x− x), (2.17)

then we want|error(x− x)|

|x− x|→ 0 (2.18)

as x→ x. The matrix J =D(f(x)) is the Jacobian matrix of f(·), which is the

matrix of its partial derivatives. Now, we inspect the eigenvalues of the Jacobian

matrix J(x) and apply the results achieved for linear systems, thus obtaining the

following results.

57


Continuous Time : Let x be a fixed point of a continuous time dynamical

system x(t) = f(x(t)), that is, f(x) = 0. The stability of a fixed point x can

be judged by the signs of the real parts of the eigenvalues of the Jacobian J(x).

If the eigenvalues of the Jacobian J(x) all have negative real part, then x is a

stable fixed point. If some eigenvalues of J(x) have positive real part, then x

is an unstable fixed point. Otherwise (all eigenvalues have nonpositive real part

and some have zero real part) we cannot judge the stability of the fixed point.

Discrete Time : Let x be a fixed point of the system x[k + 1] = f(x[k]), that

is, f(x) = x. Compute the Jacobian evaluated at x, i.e., find J(x). If the

eigenvalues of the Jacobian all have absolute value less than 1, then x is a stable

fixed point. If some eigenvalue of the Jacobian has absolute value greater than

1, then x is an unstable fixed point. Otherwise, we cannot judge the stability of

the fixed point.

Lyapunov’s Method

Linearization is a great tool for determining the stability of fixed points of

dynamical systems. Unfortunately, not always this method gives answer about

the stability of a fixed point. In the following, we introduce the Lyapunov’s

method to prove stability of dynamical systems. This method is based on the

concept of system energy. If a dynamical system models a mechanical system,

then consideration of energy is appropriate. To prove the stability of nonphysical

dynamical systems, the idea is to make up a function which behaves like the

energy, i.e., the Lyapunov function, and show that the system reduces the value

of this function (“energy”) as the time increases. Then, suppose we have a

continuous time dynamical system with state vector x, which has a fixed point

x. Let V be a function defined on the states of the space, i.e., to each state x we

assign a number V (x). Now, suppose V satisfies the following conditions:

• V is a continuously differentiable function with V (x) > 0 for all x 6= x,

and V (x) = 0.

58


• dV/dt ≤ 0 at all states x. Further, at any state x 6= x where dV/dt = 0,

the system immediately moves to a state where dV/dt < 0.

If we can find such a function V (and this can be difficult), then it must be the case

that x is a stable fixed point of the dynamical system. We know that x is stable

because as time progresses “energy” continually decreases until it bottoms out at

the fixed point. A special class of dynamical system is particularly well suited to

the Lyapunov method. These systems arise from the gradient of a function. In

particular, suppose we are given a continuous time dynamical system of the form

x(t) = f(x(t)) with fixed point x. We seek a function h(x) for which:

• f(x) = −∇h(x),

• h(x) = 0,

• h(x) > 0 for all x 6= x.

If such a function exists, then h(x) is a Lyapunov function and we may conclude

that x is stable. Indeed, as the state vector x(t) changes, the value of h(x)

decreases. The fixed points x of the system are the points where ∇h(x) = 0.

These points include the local minima of h(·), and these are precisely the stable

fixed points of the system.

Lasalle’s Invariance Principle

To conclude this section, we introduce the Lasalle’s invariance principle, which

enables one to prove asymptotic stability of an equilibrium point and will be used

in this work. We denote the solution trajectories of the system x(t) = f(x(t)) as

x(t,x0, t0), which is the solution at time t starting from x0 at t0. To introduce

Lasalle’s invariance principle, we need the definition of invariant set, which we

report in the following.

Invariant set : The set M ⊂ Rn is said to be an (positively) invariant set if

for all y ∈ M and t0 > 0, we have x(t,y, t0) ∈ M ∀t ≥ t0. It may be proved

59

2.2 Distributed Optimization

that the limit set of every trajectory is closed and invariant. We may now state

Lasalle’s Invariance Principle.

Lasalle’s Invariance Principle : Let V : Rn → R be a locally positive definite

function such that on the compact set ωc = x ∈ Rn : V (x) ≤ c we have

V (x) ≤ 0. Define

S = x ∈ ωc : V (x) = 0. (2.19)

As t→∞, the trajectory tends to the largest invariant set inside S. In particular,

if S contains no invariant sets other than x = 0, then 0 is asymptotically stable.


In this section, we consider iterative methods for the solution of a variety of

nonlinear problems. Nonlinear problems are typically solved by iterative meth-

ods, and the convergence analysis of these methods is one of the focal points

of this section. We consider descent methods, e.g., methods based on the it-

erative reduction of the cost function of an underlying optimization problem.

Throughout the section, we emphasize algorithms that are well suited for paral-

lel implementation such as Jacobi or Gauss-Seidel methods. If the optimization

problem is constrained, we also discuss gradient projection methods, which can

be naturally parallelized if the constraint set is given by the Cartesian product

of smaller sets. Then, we focus on convex constrained optimization problems,

which can be transformed into dual problems that in many cases are easier to

solve and more amenable to parallel implementations.

2.2.1 Unconstrained Optimization

We consider algorithms for minimizing in a distributed manner a continuously

differentiable cost function F (x) : Rn → R, where x = (x1, . . . ,xM ) ∈ Rn, with

xi ∈ Rni . Since the function F (·) is not supposed to be convex, iterative opti-

mization algorithms aimed at finding a solution of ∇F (x) = 0 do not guarantee

60


that such a solution is a global minimizer of F . We now recall some descent

iterative methods:

1. Jacobi algorithm : This algorithm is achieved through a simultaneous

update of the vector components of x = (x1, . . . ,xM ) as:

xi[k + 1] = xi[k]− α[Bii(x[k])]−1∇xiF (x[k]), (2.20)

i = 1, . . . ,M

where α is a positive step-size, and Bii(x[k]) is the i-th matrix element of

a positive definite block-diagonal scaling matrix. The choice Bii(x[k]) = I,

∀i, leads to the common gradient descent update, whereas Bii(x[k]) =

∇2xiF (x[k]) determines a scaled gradient update that approximates a New-

ton recursion, thus enhancing the convergence speed of the algorithm.

2. Gauss-Seidel : This algorithm sequentially updates the vector components

of x = (x1, . . . ,xM ) as:

xi[k + 1] = xi[k]− α[Bii(zi[k])]−1∇xiF (zi[k]), (2.21)

i = 1, . . . ,M

where zi[k] = (x1[k + 1], . . . ,xi−1[k + 1],xi[k], . . . ,xM [k]). Also in this

case the choice of the scaling Bii(zi[k]) leads to different Gauss-Seidel im-

plementations.

These algorithms fall in the framework of descent methods and their convergence

properties have been deeply studied in the literature [189]. Classical convergence

analysis shows that each update reduces the value of the cost function by an

amount that is bounded away from zero if the magnitude of the update is bounded

away from zero. Given a cost function that is bounded from below, it follows

that the magnitude of the update converges to zero. Then, according to this

argument, it is possible to show that∇F (x(t)) converges to zero. In the following,

we report a convergence result from [189] regarding the convergence of generic

descent methods.

61


The following assumption on the function F (·) is needed:

Assumption 1 :

a) There holds F (x) ≥ 0 (or bounded from below) for every x ∈ Rn;

b) (Lipschitz Continuity of ∇F ) The function F is continuously differentiable

and there exists a constant K such that

‖∇F (x)−∇F (y)‖2 ≤ K‖x− y‖2, ∀x,y ∈ Rn. (2.22)

Theorem 1 Suppose the Assumption 1 holds and let K1 and K2 be positive

constants. Consider the sequence generated by an algorithm of the form

x[k + 1] = x[k]− αs[k], (2.23)

where s[k] satisfies

‖s[k]‖2 ≥ K1‖∇F (x[k])‖2, ∀k, (2.24)

and

s[k]T∇F (x[k]) ≤ −K2‖s[k]‖22, ∀k. (2.25)

If 0 < α < 2K2/K, then

limk→∞

∇F (x[k]) = 0. (2.26)

Proof. The proof can be found in [189]. The proof for algorithms of the Gauss-

Siedel type follows similar arguments.

2.2.2 Constrained Optimization

Now, we consider the problem of minimizing a continuously differentiable

cost function F (x) : RL → R, in the case x = (x1, . . . ,xN ) ∈ RL lies inside a

nonempty, closed and convex set X ⊂ RL. The most used approach to solve this

problem relies on the gradient projection algorithm, which reads as follows:

x[k + 1] = ΠX [x[k]− α∇xF (x[k])] = T (x[k]), (2.27)

62


where ΠX [x] denotes the orthogonal projection of the vector x on the set X, and

T : X → X is the nonlinear mapping correspondent to the gradient projection

algorithm. Using descent arguments from [189], also in this case it is possible

to prove convergence of such algorithm to a local minimum of the constrained

problem we aim to solve. In the following, we report the convergence result.

Theorem 2 Suppose F (·) satisfies Assumption 1. If 0 < α < 2/K and if x∗ is a

limit point of the sequence x[k] generated by the gradient projection algorithm

in (2.27), then (y − x∗)T∇F (x∗) ≥ 0 for all y ∈ X. In particular, if F (·) is

convex on the set X, then x∗ minimizes F (·) over the set X.

Proof. The proof can be found in [189]. Similar results can be achieved for

scaled gradient projection algorithms.

The gradient projection algorithm is not, in general, amenable to parallel

implementation. Indeed, even if x − α∇xF (x) can be obviously parallelized as

before, the computation of the projection is, in general, a nontrivial operation

that involves all the components of x. However, in the important special case

where the constraint set X is a Cartesian product of sets Xi, where Xi is a closed

and convex set of Rni , it can be easily seen that the projection computation

can be parallelized as ΠX [x] = (ΠX1[x1] . . . ,ΠXM

[xM ]), thus leading to possible

Jacobi or Gauss-Seidel distributed versions of this algorithm.

2.2.3 Convex Constrained Optimization Problems

In the previous subsections we have analyzed some important optimization

methods that are well suited for parallel implementations, e.g., Jacobi and Gauss-

Siedel algorithms. These methods are not always applicable, e.g., where the

constraint set is not given by the Cartesian product of simpler sets. Then, we

now recall some of the basic techniques that exploit structural problem features,

enhancing in this way parallelization through suitable problem transformations.

The basic idea of these approaches is to consider a dual problem that may be

63


more suitable for parallel implementation than the original. We consider two

main cases. First, we consider a problem where the primal cost function is sepa-

rable and strictly convex. The strict differentiability of the cost function is very

important because it allows the parallelization of the dual problem. Then, we re-

call the augmented Lagrangian method, which aims to deal with the lack of strict

differentiability of the primal cost, and the consequent lack of differentiability of

the dual.

1) Separable Strictly Convex Problems : Suppose that the space RL is given

by the Cartesian product of sets RLi , where L = L1 + . . .+ LN , and consider

minN∑

i=1

Fi(xi) (2.28)

s.t. eTj x = sj, j = 1, . . . , r,

xi ∈ Pi, i = 1, . . . , N.

where Fi(xi) are strictly convex functions, xi are the components of x, ej are

given vectors in RL, sj are given scalars, and Pi are given polyhedral subsets

of RLi . The constraints eTj x = sj do not allow direct decomposition of this

problem into independent subproblems. However, a possible way to parallelize

this problem is to consider the dual problem that involves Lagrange multipliers

for these constraints. The dual problem has the form:

max q(p) (2.29)

s.t. p ∈ Rr.

where the dual function q(p) is given by

q(p) = minxi∈Pi

N∑

i=1

Fi(xi) +r∑

j=1

pj(eTj x− sj)

=N∑

i=1

qi(p)− pTs, (2.30)

where pTs =∑r

j=1 pjsj , and

qi(p) = minxi∈Pi

Fi(xi) +

r∑

j=1

pjeTjixi

, i = 1, . . . , N, (2.31)

64


with eji denoting the subvector of ej that corresponds to xi. Now, we notice

how the structure of the dual problem (2.29) is amenable to parallel computation

with separate agents computing each component qi(p) of q(p). Since the primal

cost function is strictly convex, the dual cost is continuously differentiable and

we can apply descent methods such as Jacobi or Gauss-Seidel to get the solution

in a distributed fashion. This is possible because, in contrast with the primal

problem (2.28), the dual problem (2.29) is unconstrained. Then, if the minimum

of equation (2.31) is attained at a point xi(p), the partial derivative of q with

respect to pj is given by

∂q(p)

pj= eTj x(p)− sj. (2.32)

In a distributed system where each node computes the i-th component of x, the

calculation of the dual cost gradient via (2.32) requires a single or multinode

accumulation, so that the gradient ∇q(p) can be distributed to all nodes by

means of a single or multinode broadcasting.

2) Augmented Lagrangian Method : We now recall one of the basic methods

based on a dual approach for overcoming possible lack of strict monotonicity of the

primal cost function. Consider the following constrained optimization problem:

min F (x) (2.33)

s.t. eTj x = sj, j = 1, . . . , r,

x ∈ P.

where F (x) is a convex function, ej are given vectors in RL, sj are given scalars,

and P is a given a nonempty and bounded polyhedral subset of RL. We can now

consider in place of the original problem (2.33), the following problem:

min F (x) +c

2‖Ex− s‖22 (2.34)

s.t. Ex = s,

x ∈ P.

65

2.3 Stochastic Approximation

where c is a positive scalar parameter, and Ex = s is a compact notation for the

constraints eTj x = sj, j = 1, . . . , r,. The dual problem is then

max qc(p) = infx∈P

Lc(x,p) (2.35)

s.t. p ∈ Rr.

where Lc(x,p) is the Augmented Lagrangian function:

Lc(x,p) = F (x) + pT (Ex− s) +c

2‖Ex− s‖22 (2.36)

An important method involving the augmented Lagrangian is the alternating

direction method of multipliers to minimize (2.36). It consists of successive min-

imizations of the form:

x[k + 1] = arg minx∈P

Lc[k](x,p[k]), (2.37)

followed by the updates of the vector p[k] according to

p[k + 1] = p[k] + c[k](Ex[k + 1]− s), (2.38)

where p[0] is arbitrary and c[k] is a nondecreasing sequence of positive numbers.

In [189] it is proved every limit point of the sequence x[k] is a solution of the

primal problem (2.33). One problem with the alternating method of multipliers

is that, even if the cost function F (x) is separable, the Augmented Lagrangian

is typically non separable because of the quadratic term ‖Ex − s‖22. However,

some reformulations are possible to allow parallelization of this algorithm.


Stochastic approximation methods are a family of iterative stochastic opti-

mization algorithms that attempt to find zeroes or extrema of functions that

cannot be computed directly, but only estimated via noisy observations. Mathe-

matically, this refers to solving:

minx∈E

f(x) = E[F (x, ξ)] (2.39)

66


where the objective is to find the parameter x ∈ E, which minimizes f(x) for some

unknown random variable, ξ. Denoting n as the dimension of the parameter x, we

can assume that while the domain E ⊂ Rn is known, the objective function, f(x),

cannot be computed exactly, but instead only known with an approximation.

This can be intuitively explained as follows. f(x) is the original function we

want to minimize. However, due to noise, f(x) can not be evaluated exactly. This

situation is modeled by the function F (x, ξ), where ξ represents the noise and is

a random variable. Since ξ is a random variable, so is the value of F (x, ξ). The

objective is then to minimize f(x), but through measuring F (x, ξ). A reasonable

way to do this is to minimize the expectation of F (x, ξ), i.e., E[F (x, ξ)].

The first, and prototypical, algorithms of this kind are the Robbins-Monro

[180] and Kiefer-Wolfowitz [181] algorithms, which we will report in the following

together with convergence results.

2.3.1 Robbins-Monro procedure

Let R(x) : Rn → Rn be some unknown function whose values may be mea-

sured at any point x ∈ Rn. The only information available aboutR(x) is general,

concerning, for example, continuity, monotonicity, and so on. In particular, it is

known that the equation

R(x) = 0 (2.40)

has a unique solution x∗. The problem is to determine x∗ by suitable measure-

ments of R(x). More precisely, we wish to draw up a plan for an appropriate

experiment, i.e., to specify the points x[k] ∈ Rn at which R(x) has to be mea-

sured at times k = 1, 2, . . ., in such a way x[k]→ x∗ as k →∞. The methods for

solving this problem depend on the presence of measurement errors on R(x). If

there are no errors, there are several rapidly converging methods for finding x∗,

such as Newton tangent method. If the effect of the measurement is significant,

it is in principle impossible to devise such rapidly convergent methods, and one

must employ slower procedures that we now proceed to construct. It is natural

67


to confine our attention to independent (in time) observations of R(x) and to

assume that the measurement error has zero mean and may depend on the point

at which the measurement is made. Then, if Y (k + 1,x[k], ω) is the result of a

measurement at a point x[k] at time k + 1, in the simplest case we have

Y (k + 1,x[k], ω) = R(x[k]) + Γ(k,x[k], ω) (2.41)

where Γ(k,x[k], ω) is a family of unknown zero-mean random vectors in Rn,

defined on some probability space (Ω,F ,P). The problem we have formulated

thus reduces to the determination of x∗ from the observations (2.41). In a pio-

neering paper, published in 1951, Robbins and Monro [180] proposed a recursive

procedure aimed at finding the root of R(x), given by

x[k + 1] = x[k]− α[k]Y (k + 1,x[k], ω), x[0] = x0, (2.42)

where α[k] is a sequence of iteration dependent positive step-sizes satisfying

∞∑

k=0

α[k] =∞,∞∑

k=0

α2[k] <∞. (2.43)

In Robbins and Monro [180], it was shown that if R(x) is a monotone decreas-

ing continuous bounded function and the expectations of the random variables

Γ2(k,x[k], ω) are bounded uniformly in k and x, then E‖x[k] − x∗‖ → 0, as

k → ∞, for any initial point x0 ∈ Rn. A brief explanation of the conditions in

(2.43) is given in the following. The first condition assures the numbers α[k] are

not ”too small” and is a necessary condition for the convergence of x[k] to x∗

even when there are no random errors. Indeed, if Γ(k,x[k], ω) = 0, for all k, and

the series∑∞

k=0 α[k] is convergent, then, as follows from (2.42),

∞∑

k=0

‖x[k + 1]− x[k]‖ ≤∞∑

k=0

α[k]‖R(x[k])‖ < c, (2.44)

where the constant c is independent of the initial point x0. As a consequence,

the sum of increments x[k+1]−x[k] is finite and the value of x[k] does not reach

68


x∗ as k → ∞ if, for example, the initial point x0 is sufficiently far away from

x∗. On the other hand, neither should the numbers α[k] be too large, otherwise

the random errors will prevent the convergence. It turns out that the condition∑∞

k=0 α2[k] < ∞ asymptotically damps the effect of random errors, since when

it holds we have

‖α[k]Γ(k,x[k], ω)‖ → 0, as k →∞ (2.45)

with probability one, because

E

(

∞∑

k=0

α[k]Γ(k,x[k], ω)

)2

= E

∞∑

k=0

α2[k]Γ2(k,x[k], ω) <∞.

Conditions (2.43) ensure that the step-size decays to zero, but not too fast. An

example of step-size sequence that satisfies (2.43) is

α[k] =α0

(k + 1)β, α0 > 0, 0.5 < β ≤ 1. (2.46)

Note, moreover, that according to the procedure (2.42), computation of each

consecutive point x[k + 1] requires knowledge of the preceding point x[k] only,

and there is no need for the previous experimental points (This also exemplifies

the practical value of this procedure since it does not impose exaggerated memory

capacity in computer implementation). From the mathematical standpoint, this

means that the process x[k] defined by (2.42) is a Markov process.

In the following, we will provide useful convergence results on the stochastic

procedure (2.42) from [183], first considering the case where the function R(x)

has a single zero in x∗ and then extending the result to the case of multiple

zeros. Convergence conditions are usually formulated in terms of existence of

a Lyapunov function V (x) ∈ C02 , i.e. doubly continuously differentiable with

bounded partial derivatives. We are now ready to state the first convergence

theorem on the RM stochastic approximation procedure (2.42).

Theorem 3 Let x[k]k≥0 be a Markov process defined by the RM difference

equation (2.42). Assume that there exists a function V (x) ∈ C02 satisfying the

69


conditions

V (x) > 0, V (x0) = 0, lim‖x‖→∞

V (x) = ∞ (2.47)

supǫ<|x−x0|<1/ǫ

< R(x),∇xV (x) > < 0 for ǫ > 0 (2.48)

where < ·, · > denotes the inner product operator. In addition, if there exist two

positive constants k1 and k2, such that

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ k1(1 + V (x))− k2 < R(x),∇xV (x) >, (2.49)

then, starting from an arbitrary initial condition x0, the process x[k]k≥0 con-

verges almost surely (a.s.) to x∗ as k →∞, provided the conditions (2.43) hold.

Proof. The proof can be found in [183] (Theorem 4.4.4).

Up to now, we have considered the case in which the equation R(x) = 0 has a

unique root. It is not difficult to imagine a situation in which the observer has no

such information at this disposal. Then, in the following, we report a convergence

result that generalizes the results of the previous theorem to the case where the

function R(x) has multiple zeros.

Theorem 4 Let x[k]k≥0 be a Markov process defined by the RM difference


conditions

V (x) > 0, lim‖x‖→∞

V (x) = ∞ (2.50)

supx∈Uǫ,1/ǫ(S)

< R(x),∇xV (x) > < 0 for ǫ > 0 (2.51)

where < ·, · > denotes the inner product operator, S = x : R(x) = 0 is the

solution set and Uǫ,1/ǫ(S) = x ∈ Rn : ǫ < ‖x − xs‖ < 1/ǫ,xs ∈ S, ǫ > 0. In

addition, if there exist two positive constants k, such that

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ k(1 + V (x)), (2.52)

70



verges almost surely (a.s.), as k → ∞, either to a point of the solution set

S = x : R(x) = 0, or to the boundary of its connected components, provided

that the conditions (2.43) hold.


2.3.2 Kiefer-Wolfowitz procedure

In 1952 Kiefer andWolfowitz [181], taking the Robbins-Monro method as their

point of departure, considered the following problem, involving the determination

of the maximum of an unknown function. Let f(x) : Rn → R, x ∈ Rn, be a

continuously differentiable function with a unique maximum at the point x = x∗.

Suppose that the observer may conduct independent measurements of f(x) with

a certain error, so that the measurement result Y (k + 1,x[k], ω) at a point x[k]

at the time k + 1 has the form

Y (k + 1,x[k], ω) = f(x[k]) + φ(k,x[k], ω). (2.53)

It is required to find the maximum point x∗ of f(x) or, equivalently, to solve the

system of equations ∇f(x) = 0. If the values of f(x) had been measured with

no errors, the maximum of f(x) could have been determined via the gradient

method, described by the recurrence relation

x[k + 1] = x[k] + α∇f(x[k]), (2.54)

where α is a positive constant. Nevertheless, if the errors in the measurements

of f(x) at different points are independent, the error involved in calculating

∇f(x[k]) according to these measurements will become infinitely large. The idea

of the Kiefer-Wolfowitz method is to evaluate approximate values of the gradient

in (2.54) as the quotient of the increment of the function and the increment ∆x,

setting ∆x = 2c[k]→ 0 as k →∞, at the same time “decelerating” the motion of

x[k] toward x∗, making the parameter α time dependent. The function α = α[k]

71


may be chosen in such a way that, first, the sequence x[k] does not stop too soon,

and, second, the effect of random noise is damped. It is readily seen that even

without noise the condition∞∑

k=0

α[k] =∞ (2.55)

is necessary for the solution of (2.54) to converge to x∗. To meet the second

requirement, it is frequently enough to demand that

∞∑

k=0

(

α[k]

c[k]

)2

<∞. (2.56)

According to Kiefer and Wolfowitz, at each instant k we conduct measurements

at 2l points x±i = x ± c[k]ei, where ei ∈ R

n with coordinates δij , i, j = 1, . . . , l,

and c[k] is some positive function. The approximation of ∇f(x) will then be

given by

∇f(x) ≈ ∇cf(x) =f+(c[k],x)− f−(c[k],x)

2c[k](2.57)

where f±(c[k],x) is the vector with coordinates f(x±i ), i = 1, . . . , l. Measure-

ments of the ith coordinate of the vector [f+(c[k],x)−f−(c[k],x)]/2 will involve

an error of

Γi[k + 1,x[k], ω] = [φ(k + 1,x+i , ω)− φ(k + 1,x−

i , ω)]/2. (2.58)

Thus, the Kiefer-Wolfowitz procedure for locating the maximum of f(x) is de-

scribed by the following difference equation:

x[k + 1] = x[k] +α[k]

c[k][∇cf(x) + Γ[k + 1,x[k], ω]] (2.59)

where α[k] and c[k] are certain sequence of positive numbers, and Γ[k+1,x[k], ω]

is the vector with coordinates (2.58), such that EΓ[k,x, ω] = 0. Convergence

conditions for the procedure (2.59) will be reported below.

Theorem 5 Let x[k]k≥0 be a Markov process defined by the KW difference


72


conditions

V (x) > 0, V (x0) = 0, lim‖x‖→∞

V (x) =∞, (2.60)

< ∇xf(x),∇xV (x) > < 0 for x 6= x∗, (2.61)

where < ·, · > denotes the inner product operator. In addition, if the function

f(x) has continuous partial derivatives satisfying a global Lypschitz condition,

and if there exist two positive constants k1 and k2, such that

‖∇xV (x)‖+ ‖∇xf(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ k1(1 + V (x)), (2.62)


verges almost surely (a.s.) to x∗ as k →∞, provided these conditions hold:

∞∑

k=0

α[k]c[k] <∞,∞∑

k=0

α[k] =∞,∞∑

k=0

(

α[k]

c[k]

)2

<∞, c[k] < k2. (2.63)

Proof. The proof can be found in [183].

Up to now, we have considered the case in which the function f(x) = 0 has a

unique maximum. It is not difficult to imagine a situation in which the observer

has no such information at this disposal. Then, in the following, we report a

convergence result that generalizes the results of the previous theorem to the

case where the function f(x) has multiple maxima.

Theorem 6 Let x[k]k≥0 be a Markov process defined by the KW difference


conditions

V (x) > 0, lim‖x‖→∞

V (x) = ∞ (2.64)

supx∈Uǫ,1/ǫ(S)

< ∇xf(x),∇xV (x) > < 0 for ǫ > 0 (2.65)

a[k] [< ∇cf(x)−∇xf(x),∇xV (x) >] < g[k](1 + V (x)), (2.66)

73

2.4 Graph Theory

∞∑

k=0

g[k] < ∞ (2.67)

∞∑

k=0

α[k]maxx‖∇cf(x)−∇xf(x)‖ < ∞ (2.68)

where < ·, · > denotes the inner product operator, S = x : ∇xf(x) = 0 is the

solution set and Uǫ,1/ǫ(S2) = x ∈ Rn : ǫ < ‖x − xs‖ < 1/ǫ,xs ∈ S2, ǫ > 0. In

addition, if there exist a positive constants k, such that

‖∇cf(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ k(1 + V (x)), (2.69)


verges almost surely (a.s.), as k → ∞, either to a point of the solution set

S2 = x : ∇xf(x) = 0, or to the boundary of its connected components, provided

that the following conditions hold:

∞∑

k=0

α[k] =∞,∞∑

k=0

(

α[k]

c[k]

)2

<∞, 0 < c[k] < K. (2.70)


2.4 Graph Theory

The interaction among the network nodes is properly described by a graph.

For the reader’s convenience, in this section, we briefly review the notation and

basic results of graph theory that will be used throughout this work. For the

reader interested in a more in-depth study of this field, we recommend, for ex-

ample, [82]- [86].

2.4.1 Directed Graphs: The Basic Mathematical Tool to De-

scribe Interactions

To take explicitly into account the possibility of unidirectional links among

the network nodes, we represent the information topology among the nodes by

74

2.4 Graph Theory

their (weighted) directed graph. A weighted directed graph (or digraph, for

short) G = V, E is defined as a set of nodes (or vertices) V and a set of edges

E (i.e., ordered pairs of nodes), with the convention that eij , (vi, vj) ∈ E (i.e.,

vi and vj are the head and the tail of the edge eij , respectively) means that the

information flows from vj to vi. A digraph is weighted if a positive weight aij

is associated to each edge. In our case, there are no loops, so that aii = 0. If

(vi, vj) ∈ E ⇔ (vj , vi) ∈ E , then the graph is said to be (weighted) undirected.

For any node vi ∈ V, we define the information neighbor of vi as

Ni = j = 1, . . . , N : (eij) = (vi, vj) ∈ E (2.71)

The set Ni represents the set of nodes sending data to node vi. The in-degree

and out-degree of node vi ∈ V are, respectively, defined as:

degin(vi) =

N∑

j=1

aij and degout(vi) =

N∑

j=1

aji (2.72)

Observe that for undirected graphs, degin(vi) = degout(vi). We may have the

following class of digraphs.

Balanced digraph: The node vi of a digraph G = V, E is said to be balanced

if and only if its in-degree and out-degree coincide, i.e., degin(vi) = degout(vi). A

digraph G = V, E is balanced if and only if all its nodes are balanced, i.e.,

N∑

j=1

aij =

N∑

j=1

aji, ∀i = 1, . . . , N. (2.73)

Path/Cicle: A strong path in a digraph G is a sequence of distinct nodes

v1, v2, ..., vq ∈ V such that (vi−1, vi) ∈ E , for i = 2, . . . , q. If v1 = vq, the path is

said to be closed. A weak path is a sequence of distinct nodes v1, v2, . . . , vq ∈ V

such that either (vi−1, vi) ∈ E or (vi, vi−1) ∈ E , for i = 2, . . . , q. A strong cycle

(or circuit) is a closed strong path.

Directed Tree/Forest: A digraph with N nodes is a (rooted) directed tree if

it has N − 1 edges and there exists a distinguished node, called the root node,

75

2.4 Graph Theory

which can reach all the other nodes through a (unique) strong path. Hence, a

directed tree cannot have cycles and every node, except the root, has one and

only one incoming edge. A digraph is a (directed) forest if it consists of one or

more directed trees. A subgraph Gs = Vs, Es of a digraph G, with Vs ⊆ V and

Es ⊆ E , is a directed spanning tree (or a spanning forest) if it is a directed tree

(or a directed forest) and it has the same node set as G; i.e., Vs = V. We say

that a digraph G contains a spanning tree (or a spanning forest) if there exists a

subgraph of G that is a directed spanning tree (or a spanning forest).

Connectivity: In a digraph there are many degrees of connectedness:

1) a digraph is strongly connected (SC) if any ordered pair of distinct nodes can

be joined by a strong path;

2) a digraph is quasi strongly connected (QSC) if, for every ordered pair of nodes

vi and vj, there exists a node r that can reach either vi or vj via a strong path;

3) a digraph is weakly connected (WC) if any ordered pair of distinct nodes can

be joined by a weak path;

4) a digraph is disconnected if it is not weakly connected.

According to the above definitions, it is straightforward to see that strong connec-

tivity implies quasi strong connectivity and that quasi strong connectivity implies

weak connectivity, but the converse, in general, does not hold. For undirected

graphs, instead, the above notions of connectivity are equivalent: An undirected

graph is connected if any two distinct nodes can be joined by a path. Moreover,

it is easy to check that the quasi strong connectivity of a digraph is equivalent to

the existence of a directed spanning tree in the graph.

Condensation Digraph: When a digraph G is WC, it may still contain strongly

connected subgraphs. A maximal subgraph of G, which is also SC, is called a

strongly connected component (SCC) of G. Since a node is SC, it follows that

76

2.4 Graph Theory

every node lies in an SCC. Using this concept, any digraph G can be partitioned

into SCCs, let us say G1 = V1, E1, . . . , GK = VK , EK, where Vk ⊆ V and

Ek ⊆ E , k = 1, . . . ,K, denote the set of nodes and edges lying in the k-th SCC,

respectively.

The connectivity properties of a digraph may be better understood by referring

to its corresponding condensation digraph. We may reduce the original digraph

G to the condensation digraph G∗ = V∗, E∗ by associating the node set Vk of

each SCC Gk of G to a single distinct node v∗k ∈ V∗k of G∗ and introducing an

edge in G∗ from v∗i to v∗j if and only if there exists some edges from the SCC Gi

and the SCC Gj of the original graph. An SCC that is reduced to the root of a

directed spanning tree of the condensation digraph is called root SCC (RSCC).

Observe that, by definition, the condensation digraph has no cycles. Building on

this property, we have the following.

Lemma 1 Let G∗ = V∗, E∗ be the condensation digraph of G , composed by K

nodes. Then, the nodes of G∗ can always be ordered as v∗1, . . . , v∗K ∈ V

∗, so that

the existing edges in G∗ are in the form

(v∗i , v∗j ) ∈ E

∗, with 1 ≤ j < i ≤ K, (2.74)

where v∗1 has zero in-degree.

The ordering v∗1 , . . . , v∗K satisfying (2.74) can be obtained by the following iter-

ative procedure. Starting from v∗1 , remove v∗1 and all its out-coming edges from

G∗. Since the reduced digraph with K − 1 nodes has no (strong) cycles by con-

struction, there must exist a node with zero in-degree in it. Let us denote such

a node by v∗2 . Then, no edges in the form (v∗2 , v∗j ), with j > 2, can exist in the

reduced digraph (and thus in G∗). This justifies (2.74) for i = 2 and j = 1, 2.

The rest of (2.74), for j > 2, is obtained by repeating the same procedure on

the remaining nodes. The connectivity properties of a digraph are related to the

structure of its condensation digraph, as given in the following Lemma (we omit

the proof because of space limitations).

77

2.4 Graph Theory

Figure 2.2: Examples of graphs: (a) Strongly connected graph. (b) Quasi strongly

connected graph with one root strongly connected component and two strongly

connected components. (c) WC graph containing a two-tree forest.

Lemma 2 Let G∗ be the condensation digraph of G . Then: i) G is SC if and

only if G∗ is composed by a single node; ii) G is QSC if and only if G∗ contains

a spanning directed tree; iii) if G is WC, then G∗ contains either a spanning

directed tree or a (weakly) connected directed forest.

The concept of condensation digraph is useful to understand the network syn-

chronization behaviors [135] and leadership problems in coordinated multi-agent

systems [114]. Some examples of graph topologies are shown in Figure 2.2, where

we report three topologies, namely: (a) an SC digraph, (b) a QSC digraph with

three SCCs, and (c) a WC (not QSC) digraph with a two-tree forest. For each

digraph, we also sketch its decomposition into SCCs and RSCCs.

2.4.2 Algebraic Graph Theory

We recall now some basic relationships between the connectivity properties of

the digraph and the spectral properties of the matrices associated to the digraph,

since they play a fundamental role in the stability analysis of the system proposed

in this work. In the following, we denote by 1M and 0M the M -length column

78

2.4 Graph Theory

vector of all ones and zeros, respectively. Given a digraph G = V, E, we

introduce the following matrices associated to G :

• the M ×M adjacency matrix A is composed of entries [A]ij = aij, i, j =

1, . . . ,M , equal to the weight associated with the edge eij , if eij ∈ E , or

equal to zero, otherwise;

• The degree matrix D is a diagonal matrix with diagonal entries [D]ii =

degin(vi);

• The (weighted) Laplacian L is defined as

∑Mk 6=i=1 aik, if j = i;

−aij , if j 6= i.(2.75)

Using the adjacency matrix A and the degree matrix D, the Laplacian L

can be rewritten in compact form as L =D −A.

By definition, the Laplacian matrix L in (2.75) has the following properties:

i) it is a diagonally dominant matrix; ii) it has zero row sums; and iii) it has

nonnegative diagonal elements. From i)-iii), invoking Gersgorin’s disk Theorem

[178], we have that zero is an eigenvalue of L corresponding to a right eigenvector

in the NullL ⊇ span1M, i.e.,

L1M = 0M (2.76)

while all the other eigenvalues have positive real part. This also means that

rank(L) = M − 1. Moreover, from (23) and (26), it turns out that balanced

digraphs can be equivalently characterized in terms of the Laplacian matrix L:

A digraph is balanced if and only if 1M is also a left eigenvector of L associated

with the zero eigenvalue, i.e.,

1TML = 0TM (2.77)

or equivalently 12(L+LT ) is positive semidefinite.

The relationship between the connectivity properties of a digraph and the

spectral properties of its Laplacian matrix are given by the following.

79

2.4 Graph Theory

Lemma 3 Let G = V, E be a digraph with Laplacian matrix L. The multiplic-

ity of the zero eigenvalue of L is equal to the minimum number of directed trees

contained in a spanning directed forest of G.

Corollary 1 The zero eigenvalue of L is simple if and only if G contains a

spanning directed tree (or, equivalently, it is QSC).

Observe that, since the strong connectivity of the digraph implies QSC, the re-

sults provided for SC digraphs, can be obtained as special case of Corollary 2.

Specifically, we have the following.

Corollary 2 Let G = V, E be a digraph with Laplacian matrix L. If G is SC,

then L has a simple zero eigenvalue and a positive left-eigenvector γ associated

to the zero eigenvalue.

According to Corollary 2, because of (2.76), the Laplacian of a QSC digraph has

an isolated eigenvalue equal to zero, corresponding to a right eigenvector in the

span1M. Observe that, for undirected graphs, Corollary 3 can be stated as

follows: rank(L) = M − 1 if and only if G is connected. For directed graphs,

instead, the only if part does not hold. We describe now the structure of the

left-eigenvector γ of the Laplacian matrix L associated to the zero eigenvalue, as

a function of the network topology. We have the following.

Lemma 4 [135] Let G = V, E be a digraph withM nodes and Laplacian matrix

L. Assume that G is QSC with K SCC’s G1 = V1, E1, . . . , GK = VK , EK,

with Vi ⊆ V, Ei ⊆ E, |Vi| = ri and∑

i ri = M , numbered w.l.o.g. so that G1

coincides with the RSCC of G. Then, the left-eigenvector γ = [γ1, . . . , γM ]T of L

associated to the zero eigenvalue has the following structure

γi =

> 0, iff vi ∈ V;

= 0, otherwise.(2.78)

If G1 is also balanced, then γr1 = [γ1, . . . , γr1 ]T ∈ span1r1, where r1 , |V1|.

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2.4 Graph Theory

Undirected graphs

In this section, we focus on undirected graphs and the spectral properties of

the associated matrices. Let model the interaction among the network nodes as

an undirected graph G = (V, E), where V = 1, 2, ...,M denotes the set of nodes

and E ⊆ V × V is the edge set. The structure of the graph is described by the

symmetric M ×M adjacency matrix A := aij, whose entries aij are either

positive or zero, if there is or not a link between nodes i and j, respectively.

Properties : Let G = (V, E) be an undirected graph of order M with a symmetric

non-negative adjacency matrix A = AT . Then, these statements hold true:

1. L is a positive semidefinite matrix that satisfies the following sum-of-squares

(SOS) property

xTLx =1

2

M∑

i=1

M∑

j=1

aij(xj − xi)2, x ∈ R

n; (2.79)

2. The graph has c ≥ 1 connected components iff rank(L) = M − c. In

particular, G is connected iff rank(L) =M − 1;

3. Let G be a connected graph, then for any x such that x ⊥ 1M , we have 1

0 < λ2(L) ≤xTLx

‖x‖2≤ λM (L). (2.80)

The quantity λ2(L) is known as the algebraic connectivity of the graph and is a

measure of performance/speed of consensus algorithms [132]. If the symmetric

graph modeling the interaction among the nodes is connected, the multiplicity

of the null eigenvalue of the Laplacian is one. In particular, we consider M -

dimensional graph Laplacians defined by

L = L⊗ In (2.81)

1We recall that, for a connected graph, the nullspace of L has dimension 1 and it is spanned

by the vector 1. Hence, a vector x ⊥ 1M indicates a vector lying in a subspace orthogonal to

the nullspace of L.

81

2.4 Graph Theory

where ⊗ denotes Kronecker product. This multidimensional Laplacian satisfies

the following property:

xT Lx =1

2

M∑

i=1

M∑

j=1

aij‖xj − xi‖2, x ∈ R

nM . (2.82)

where x := (xT1 , . . . ,x

TM )T and xi ∈ R

n. Furthermore, the spectrum of the

multidimensional Laplacian matrix is such that

λ(L) = λ(L)⊗ 1n (2.83)

where λ(L) = [λ1(L), . . . , λM (L)] ∈ RM is the vector containing the eigenvalues

of L.

82

Chapter 3

Distributed Resource

Allocation Based on Swarming

Mechanisms

The goal of this chapter is to propose a bio-inspired radio access mechanism for

cognitive networks mimicking the behavior of a flock of birds swarming in search

for food in a cohesive fashion without colliding with each other. The equivalence

between swarming and radio resource allocation is established by modeling the

interference distribution in the resource domain, e.g. frequency and time, as the

spatial distribution of food, while the position of the single bird represents the

radio resource chosen by each radio node. The swarming mechanism is enforced

by letting every node allocate its resources (power/bits) in the time-frequency re-

gions where the interference is minimum (the food density is maximum), avoiding

collisions with other nodes (birds), yet limiting the spread in the time-frequency

domain (i.e., maintaining the swarm cohesion). The solution is given as the dis-

tributed minimization of a functional, borrowed from social foraging swarming

models, containing the average interference plus repulsion and attraction terms

that help to avoid conflicts and maintain cohesiveness, respectively.

83

3.1 Introduction on Cognitive Radio and Dynamic Radio Access

3.1 Introduction on Cognitive Radio and Dynamic

Radio Access

The radio frequency spectrum is a scarce natural resource and its efficient use

is of the utmost importance. The spectrum bands are usually licensed to certain

services, such as mobile, fixed, broadcast, and satellite, to avoid harmful interfer-

ence between different networks and users. Most spectrum bands are allocated

to certain services but worldwide spectrum occupancy measurements show that

only portions of the spectrum band are fully used. Moreover, there are large

temporal and spatial variations in the spectrum occupancy. In the development

of future wireless systems the spectrum utilization functionalities will play a key

role due to the scarcity of unallocated spectrum. Moreover, the trend in wireless

communication systems is going from fully centralized systems into the direction

of self-organizing systems where individual nodes can instantaneously establish

ad-hoc networks whose structure is changing over time. Cognitive radios [2],

with the capabilities to sense the operating environment, learn and adapt in real

time according to environment creating a form of mesh network, are seen as a

promising technology. Cognitive radio is an intelligent wireless communication

system that is aware of its surrounding environment, learns from the environ-

ment and adapts its internal states to statistical variations in the incoming RF

stimuli by making corresponding changes in certain operating parameters in real

time [3]. The primary objectives of the cognitive radio are to provide highly

reliable communications whenever and wherever needed and to utilize the radio

spectrum efficiently. The key issues in the cognitive radio are awareness, intel-

ligence, learning, adaptivity, reliability, and efficiency. The term cognitive radio

was first suggested by Mitola [2]. He defines the cognitive radio as a radio driven

by a large store of a priori knowledge, searching out by reasoning ways to deliver

the service the users want. The cognitive radio is reconfigurable and built on the

software-defined radio (SDR). The aim of the cognitive radio is to use the natural

resources efficiently including frequency, time, and transmitted energy. Spectral

84


efficiency is playing an increasingly important role as future wireless communica-

tion systems will accommodate more and more users and high performance (e.g.

broadband) services. Cognitive radio technologies can be used in lower priority

secondary systems that improve spectral efficiency by sensing the environment

and then filling the discovered gaps of unused licensed spectrum with their own

transmissions [2,3]. The opportunistic access of the radio spectrum by unlicensed

users is a problem that is attracting a large interest in the research community

as well as in the industry sector, as a way to improve the efficiency thanks to

a dynamic radio resource allocation as opposed to conventional rigid spectrum

access [2,3]. Transmission techniques for cognitive radio systems include overlay,

underlay and interweave [4]. Underlay or interference avoidance model allows con-

current transmission of primary and secondary users in ultra wideband (UWB)

fashion where the primary users are protected by enforcing spectral masks on the

secondary signals so that the generated interference is below the noise floor for

the primary user. However, underlay allows only short-range communication due

to the power constraints. Overlay or known interference model also allows con-

current transmission of primary and secondary users. The secondary users use

part of their transmission power for relaying the data of primary users and part

of the power for their own secondary transmission. In the interweave model the

cognitive radio monitors the radio spectrum periodically and opportunistically

communicates over the spectrum holes. The three major tasks of the cognitive

radio include [3]:

1. radio-scene analysis,

2. channel identification, and

3. dynamic spectrum management and transmit-power control.

The radio-scene analysis includes the detection of spectrum holes by for example

sensing the radio frequency spectrum. The channel identification includes estima-

tion of the channel state information which is needed at the receiver for coherent

85


detection. The transmitter power control and dynamic spectrum management

select the transmission power levels and frequency holes for transmission based

on the results of radio scene analysis and channel identification. The first two

tasks are usually carried out in the receiver (RX) while the third task is carried

out in the transmitter (TX), which requires some form of feedback from the RX.

The cognitive radio approach can be extended to cognitive networks. A cogni-

tive radio network is an intelligent multiuser wireless communication system that

perceives the radio-scene, adapts to variations in the environment, facilitates

communication between users by cooperation, and controls the communication

through proper allocation of resources [3]. The cognitive network encompasses

a cognitive process that can perceive current network conditions, and then plan,

decide, and act on those conditions [5]. The network can learn from adaptations

and use them to make future decisions taking into account end-to-end goals.

Cognitive networks require a software adaptable network to implement the ac-

tual network functionality and allow the cognitive process to adapt the network.

The basic idea in cognitive networks is to have a hierarchical structure where

unlicensed users, also known as secondary users (SU’s), are allowed to use tem-

porally unoccupied communication resources, like frequency bands, time slots or

user codes, under the constraint of not interfering (or producing a tolerable in-

terference) towards licensed, or primary, users. A key step in cognitive networks

is the ability of the opportunistic users to sense the resource domains, either

time slots or frequency subchannels, and then use the unoccupied resources un-

til possible and release them as soon as primary users access them. A broad

survey on cognitive radios is [92], whereas a more specific survey on dynamic

spectrum access methods is given in [93]. Besides cognitive radios, another in-

teresting area of application of dynamic radio access is femtocell networks, where

a potential massive deployment of femto-access points can determine an intoler-

able interference towards macrocell station users. In this case, the high number

of femto-access points demands for decentralized radio access strategies, aided

with proper channel sensing. Dynamic access based on sensing has been studied

86


in a series of works, see e.g. [95–101, 104–111]. In all these works it has been

emphasized how the sensing and access strategies should be designed jointly to

optimize the system performance. In particular, the authors of [100] show how

to maximize the throughput of a secondary user considering the identification of

spectral opportunities, the sensing strategy and the access strategy jointly. The

primary user activity is modeled as a discrete-time Markov chain and the effect

of channel estimation errors is taken into account. Discrete-time Markov models

assume some kind of synchronization between primary and secondary users. In

situations where this synchronization cannot be taken for granted, as in WLAN

for instance, a continuous-time Markov process is assumed to model the primary

user activity, as in [99]. The proposed methods do not prevent collisions, but try

to maximize throughput under a collision constraint. In [102] it is shown that,

when the collision constraint is tight, the strategy can be implemented with a

simple memoryless policy with periodic channel sensing. The previous works con-

centrate on the decisions about accessing the available channels or not. In [104]

it was shown how to determine the decision thresholds in order to maximize

the opportunistic throughput, in a multicarrier setup, for a given set of rates

over the available subcarriers. In [106] it was then shown, in the same multi-

carrier framework, that a substantial performance improvement can be obtained

by choosing decision thresholds and power allocation jointly, rather than sepa-

rately. Most works concentrate on the access from a single secondary user into

a system partially occupied by primary users, whereas relatively fewer papers

address the uncoordinated access from multiple secondary users. The multiuser

case was specifically addressed in [103]. Game-theoretic approaches have also

been proposed, as a way to derive decentralized access strategies. In particular,

in [107] the multiuser access problem was formulated as a game whose players are

the secondary nodes, who aim at maximizing their rate under the constraint of

inducing no interference at all, or only limited interference, to the primary users.

The joint optimization of detection thresholds and power allocation in a multi-

ple secondary user scenario was addressed in [111]. In [107,111], every cognitive

87


user was assumed to have perfect knowledge of the aggregated interference from

primary users. In practice, this is a rather idealistic assumption. A Bayesian

game-theoretic approach was considered in [108–110].

In this work, we follow a rather alternative path and, inspired by biological

models, we formulate the search for radio resources, i.e. time and frequency slots,

as the search for food by a flock of birds swarming in a cooperative manner, but

without any centralized control. The equivalence between the two problems is

the following. The interference distribution in the time-frequency plane takes

the role of the food spatial distribution: The birds (radio nodes) fly (allocate

their resources) over the regions (time-frequency domain) where there is more

food (less interference). During the flight, the birds move (choose their time-

frequency slots) in a coordinated way, even in the absence of any central control, in

order to avoid collisions (conflicts over common radio resources), yet maintaining

the swarm cohesion (i.e., avoiding unnecessary spread in the occupancy of the

time-frequency plane). Inspired by the swarm models proposed by Gazi and

Passino in [117, 118], properly modified according to our goals, we show how

the decentralized resource allocation can be formulated as the minimization of a

proper functional including the interference distribution over the radio resource

plane, plus the combination of a repulsion and an attraction term, introduced

to avoid conflicts over common resources while preventing, at the same time, an

excessive spread in the resource domain.

The basic contributions of this chapter are the following:

1) we propose the application of swarming mechanisms to radio resource alloca-

tion in cognitive radios;

2) we provide upper and lower bounds on the spread of the swarm, as a function

of the swarm connectivity;

3) we propose fast versions of the swarming algorithm, useful for our application,

and we apply such procedures to the dynamic resource allocation in the frequency

domain;

88


4) we apply the proposed procedure to the case where the primary users in a

cognitive radio are modeled as statistically independent homogeneous continuous-

time Markov processes;

5) we extend the swarming algorithms to the case of inter-nodes communications

affected by random disturbances;

6) we derive the convergence properties of the proposed algorithms in the presence

of random disturbances such as link failures, quantization and estimation errors.

The chapter is organized as follows. Section 3.2 describes the swarm model and

formulates the search of available time/frequency slots as the distributed mini-

mization of a global potential function. In Section 3.3 and 3.4, we introduce a

continuous-time swarm model, analyzing the cohesiveness of the swarm in case

of local interactions among the nodes and providing closed form expressions for

the upper and lower bounds of the swarm size. This analysis is useful to capture

the effect of the network topology and of the swarm parameters on the spread

over the resource domain. In Section 3.5, we study in more detail the swarming

mechanism in a one-dimensional domain, e.g. the frequency domain. We provide

first a local stability analysis useful to show that the introduction of the attrac-

tion and unbounded repulsion terms in the functional to be minimized does not

affect the stability of the system. Then, we propose fast swarming methods based

on a proper selection of the descent direction of a scaled gradient optimization.

The first method is an approximation of a Newton based optimization and im-

proves the convergence speed of the algorithm; the second method adapts the

swarming speed with respect to the interference power perceived by the swarm.

Numerical examples show the main algorithm’s features. We provide also a nu-

merical validation of our theoretical findings about the spread of the swarm, as

a function of the main system parameters. In Section 3.6, we describe the appli-

cation of the proposed model to the distributed resource allocation problem on

a time-frequency plane. We consider both a static interference scenario, where

the interference activity is assumed to be known and constant along the dura-

89

3.2 Problem Statement

tion of the swarming algorithm, and a dynamic interference scenario. In this

latter case, the interference activity over each frequency subchannel is modeled

as a continuous-time Markov chain. Section 3.7 describes the discrete-time im-

plementation of the proposed swarming procedure, giving convergence results for

unprojected and projected swarming methods. Finally, in Section 3.8 we ex-

tend the swarm-based resource assignment mechanism to the more realistic case

where the packets exchanged among the cognitive nodes, while running the dis-

tributed assignment mechanism, are randomly dropped and the transmitted data

are encoded with a finite number of bits. Then, using stochastic approximation

arguments, we derive the convergence properties of the proposed algorithms in

the presence of random disturbances such as link failures, quantization noise and

estimation errors. Several simulation examples are also given in order to corrob-

orate the theoretical results and show the effect of the radio channel impairments

on the performance of the proposed algorithms.


The problem we wish to solve is the assignment of time and/or frequency

slots to cognitive (or femto) users in order to minimize interference towards pri-

mary (or macrocell) users and avoid conflicts among the cognitive users, while

keeping the spread in frequency and time as reduced as possible. A centralized

controller knowing the spatial distribution of the primary users’ activity in the

time-frequency domain, as perceived by each cognitive node, could solve this

non-trivial assignment problem. However, besides the computational complexity

aspects, a centralized approach would require considerable signaling between the

secondary nodes and the controller. Furthermore, since femto-access points are

owner-operated devices, they are not necessarily under the control of a central

authority. It is then of interest to examine decentralized resource assignment

techniques. We formulate the problem as follows. Let us consider now a set of

M secondary users whose goal is to allocate resources (bits/power) dynamically

90


in a domain typically occupied by primary users, with the aim of minimizing the

interference towards the primary users. A typical setting is the case where the

resource space is a time-frequency frame, as in the 4G mobile communication

standard LTE [187]. In such a case, the goal of every secondary user is to find

out a time slot and/or a frequency subchannel, temporally unoccupied by other

users. The problem arises when the number of secondary users is very high and

there is no central authority assigning the resources on demand. In such a case,

it is necessary to devise a decentralized mechanism to assign resources under the

constraint of keeping the interference towards primary and secondary users as

low as possible. To have a notation as general as possible, we denote by n the

dimension of the resource domain and the single resource selected by node i is

described by a vector xi ∈ Rn, whose entries denote, for example, a frequency

subchannel and a time slot (n = 2 in this case). In the presence of many SUs,

the problem is how to access time and/or frequency slots that are vacant, while

at the same time avoiding conflicts between SUs, without requiring a centralized

coordination node. To avoid conflicts, we propose an iterative algorithm where,

at each iteration, every node broadcasts the vector that is planning to occupy

to its nearest neighbors. The interaction among the SU nodes is modeled as an

undirected graph G = (V,E). We assume that there is a link (edge) between two

nodes if the distance between them is less than a prescribed value (the coverage

radius), dictated by the node’s transmit power and the radio channel characteris-

tics. The graph of the network topology can be described by the adjacency matrix

A := akl, composed of nonnegative entries aij ≥ 0, the degree diagonal matrix

D, whose diagonal entries are dii :=∑M

l=1 aij , and the Laplacian L, defined as

L =D −A. The set of neighbors of a node i is Nk, defined as

Ni = l ∈ V : aij > 0. (3.1)

Node k communicates (and interferes) with node j if i is a neighbor of i (or

aij > 0). We denote by Ii(xi) ∈ C1 : Rn×R→ R the interference power over the

slot having coordinate vector xi (e.g., a frequency subchannel or a time-frequency

91


slot) perceived by node i. The goal of node i is to select the time-frequency slot,

having coordinates xi, where Ii(xi, t) is at a minimum. At the same time, each

node wants to prevent conflicts with the other SUs, while avoiding an excessive

spread over the resource domain.

The resource allocation problem can then be formulated mathematically as

the search of the resource vector x =[

xT1 , . . . ,x

TM

]T∈ R

nM , from the whole

population of cognitive nodes, that minimizes in a distributed fashion the global

potential function:

J(x) =M∑

i=1

Ii(xi) +1

2

M∑

i=1

M∑

j=1

aij[Ja(‖xj − xi‖)− Jr(‖xj − xi‖)], (3.2)

whose first term∑M

i=1 Ii(xi) represents the overall interference power over the

optimization domain (e.g., the time-frequency plane) perceived by the swarm,

while the second and third term are two penalty terms taking into account,

respectively the spread of resources and the collisions, in the resource domain.

Neglecting for a moment the second term on the right-hand side (RHS) of (3.2),

the minimization of (3.2) leads every node to find a position xi such that the

overall interference power, as perceived by the swarm, is minimum. This is a way

to let the SUs to fill the gaps in the time-frequency domain. However, such a

solution would not prevent two different SUs to choose the same position, thus

conflicting with each other. Actually, if the sum of interference functions Ii(xi)

had a common single minimum, every node would tend to occupy the same value,

thus inducing an overall conflict among SUs. To avoid this situation, the second

term on the RHS of (3.2) contains a repulsion function 12

∑Mi=1

∑Mj=1 aijJr(‖xj −

xi‖) that is maximum when two nodes occupy the same position (i.e., xj = xi).

Hence, the purpose of the repulsion term is to avoid collisions. However, the

repulsion term alone might lead to an excessive dispersion of the slots occupied

by the whole set of nodes, in the resource domain. This is also undesirable,

because it may imply the occupation of an unjustified large region in the spectral

domain or of distant time slots in the time domain, thus running into a non-

92


stationarity problem. To avoid excessive dispersion, we introduce an attraction

function 12

∑Mi=1

∑Mj=1 aijJa(‖xj −xi‖) that is minimum when the vectors xi are

all close to each other. Hence, in summary, the attraction and repulsion terms

in (3.2) are chosen so that the overall system tends to remain cohesive, without

creating conflicts. Furthermore, there is a unique distance at which the attraction

and repulsion forces balance: the so called equilibrium distance in the biological

literature [130], [131]. This distance, in our case, is related to the bandwidth of

the frequency slot or the duration of the time interval.

Remark: Function (3.2) is reminiscent of the social foraging function, introduced

by Gazi and Passino in [118] and subsequently generalized in [119]. The objective

of [118] and [119] was to model the behavior of a swarm of birds searching for food,

while moving collectively as a swarm and yet avoiding collisions. In their case,

the three terms represented, respectively, the spatial distribution of food, the

attraction and repulsion forces among the birds. In [118], the interaction among

birds was modeled as a fully connected graph, i.e. aij = 1,∀i, j. The model was

then generalized in [119] to deal with an asymmetric graph. In our work, the

coefficients aij take into account the existence of a radio link between two nodes

and then they depend on radio channel characteristics, more specifically, on the

physical distance between the nodes i and j. In our set-up, the coefficients aij may

assume any real non-negative value, but because of the radio channel reciprocity,

they satisfy the symmetry condition aij = aji. This simplifies our analysis with

respect to [119], yet providing more general results than [118]. In particular,

both [118] and [119] concentrated on the swarm cohesiveness, or stability. The

main goal of their analysis was then to provide an upper bound for the spatial

spread of the swarm. Conversely, in our application, we are concerned with two

main issues, collisions and spread: We want to limit the spread in the resource

domain and avoid, or limit as much as possible, collisions, especially between

nearby nodes. For this reason, in the next section, the goal of our analysis is

to provide both a lower and an upper bound of the swarm, to be able to assess

93

3.3 Continuous-Time Distributed Optimization

the swarm properties in terms of overall occupancy and collision. Interestingly,

thanks to the symmetry of our graph, we will provide simple expressions for the

lower and upper bounds which yield physical insight about the role played by the

graph connectivity. Before starting our analysis, it is worth to point out that the

function (3.2), while mathematically similar to the function studied in [118], [119],

it is functionally different. In fact, in [118], [119], the vectors xi indicate a spatial

position and the coefficients aij are also related to the spatial positions of nodes

i and j. Conversely, in our case, the vectors xi indicate the resource domain,

i.e. time/frequency, while the coefficients aij are related to the spatial domain,

as they depend on the spatial distance between the nodes. We will see that this

makes a significant difference in the final result because it enables spatial reuse

of radio resources. For the moment, we consider, for simplicity, time-invariant

coefficients. Since in our application the exchange of information among nodes

occurs over wireless channels, in the following we will analyze the case where the

coefficients aij are random, in order to incorporate channel fading phenomena,

quantization and noise.


Our goal in this work is the distributed minimization of (3.2). A possible way

to achieve the solution in decentralized form is to use a simple gradient based

optimization, so that every node starts with an initial guess, let us say xi(0), and

then it updates its own resource allocation vector xi(t) in time according to the

following dynamical system:

xi(t) = −∇xiJ(x(t))

= −∇xiIi(xi(t)) +

M∑

j=1

aij g(xj(t)− xi(t)), (3.3)

i = 1, . . . ,M , with g(·) denoting a vector function defined as

g(xj − xi) = [ga(‖xj − xi‖)− gr(‖xj − xi‖)](xj − xi), (3.4)

94


where

ga(‖xj − xi‖)(xj − xi) =1

2∇xiJa(‖xj − xi‖), (3.5)

gr(‖xj − xi‖)(xj − xi) =1

2∇xiJr(‖xj − xi‖). (3.6)

There are several ways to choose the function in (3.4). In this work we consider

a constant attraction term, i.e.

ga(‖xj − xi‖) = cA cA > 0, (3.7)

and unbounded repulsion, i.e.

gr(‖xj − xi‖) =cR

‖xj − xi‖2, cR > 0, ∀ ‖xj − xi‖, (3.8)

or bounded repulsion, i.e.

gr(‖xj − xi‖)‖xj − xi‖ ≤ cR, cR > 0, ∀ ‖xj − xi‖. (3.9)

An example of repulsion function satisfying the boundness assumption in (3.9) is

given by:

gr(‖xj − xi‖) = cR exp

(

−‖xj(t)− xi(t)‖

2

cG

)

, (3.10)

cR, cG > 0, ∀ ‖xj − xi‖.

Examples of the resulting attraction and repulsion behaviors are shown in Fig.

3.1 and 3.2. These choices are instrumental to endow the system with the desired

behavior and they are simple enough to allow for mathematical tractability. The

constant attraction term in (3.7) determines an intensity of the attraction force in

(3.4) that is directly proportional to the distance between resources. Unbounded

repulsion is appealing in our intended application as it prevents collisions among

nodes and ensures the existence of a lower bound on the swarm size, as we will

see in the following section. At the same time, the bounded repulsion behavior

in (3.10) makes the coupling function in (3.4) continuously differentiable.

95


−5 0 5−10

−8

−6

−4

−2

0

2

4

6

8

10

Distance between the individuals

Ma

gn

itu

de

Figure 3.1: Magnitude of the coupling function g(·) in (3.4) with linear attraction

(3.7) and unbounded repulsion (3.8), using the values cA = 1 and cR = 2. The

distance between the red points and zero is the equilibrium distance between the

swarm agents.

The parameters of the functions ga(·) and gr(·) are chosen so that, at large

distances (in the resource domain), the attraction term dominates, while at short

distances is the repulsion term to dominate, and there is a unique distance where

attraction and repulsion balance. The red dots in Fig. 3.1 and 3.2 show examples

of equilibrium distance between swarm resources considering the unbounded and

bounded repulsion functions. In our setting, this equilibrium distance is chosen

proportional to the bandwidth of the frequency slot, in the frequency domain, or

to the duration of the elementary time slot and can be adjusted acting on the

swarm parameters cA and cR. It is important to remark, about the updating rule

(3.3), that each individual in the swarm has to estimate only local parameters:

the gradient of the interference level, evaluated only on its intended running

position xi, and the balance of attraction and repulsion forces with its neighbors.

96

3.4 Stability and Cohesion Analysis

−5 0 5−8

−6

−4

−2

0

2

4

6

8

Distance between the agents

Ma

gn

itu

de

Figure 3.2: Magnitude of the coupling function g(·) in (3.4) with linear attraction

(3.7) and bounded repulsion (3.10), using the values cA = 1, cR = 10 and cG = 2.

The distance between the red points and zero is the equilibrium distance between

the swarm agents.


Before studying the stability of the swarm (3.3), it is useful to analyze the

motion of the swarm center: x = 1/M∑M

i=1 xi. The trajectory of the center is

given by:

˙x = −1

M

M∑

i=1

∇xiIi(xi) +1

M

M∑

i=1

M∑

j=1

aij g(xj − xi)

= −1

M

M∑

i=1

∇xiIi(xi), (3.11)

where the equality of the second term of the RHS of (3.11) to zero follows from

the symmetry condition aij = aji and from the fact that g(·) is an odd function.

The above equation means that the center of the swarm moves until the agents

97


reach a position where the average gradient is zero. The fact that is the average

gradient to determine the motion of the swarm center, rather than the individual

gradients, is appealing in our context because the averaging operation reduces the

effect of undesired zero-mean fluctuations due to observation noise or to errors in

the estimate of the gradient.

3.4.1 Profiles with Bounded Gradient

We will now analyze the cohesiveness of the swarm under some assumptions

on the attraction/repulsion functions and on the interference profile. To this

end, we define the displacement vector between the position xi of node i and the

center of the swarm as ∆i = xi − x. Deriving bounds on the magnitude of the

vector ∆ = [∆1T , . . . ,∆M

T ]T is useful to quantify the size of the swarm and

then, ultimately, the spread in the resource allocation domain. The assumptions

needed for our derivations are the following.

Assumption A.1 : The interference profile functions Ii(y) ∈ C1 and there exists

a constant σ > 0 such that

‖∇yIi(y)‖ ≤ σ, ∀ i,y. (3.12)

This assumption is quite general and it only requires the gradient of the profile to

be bounded. This hypothesis is indeed very reasonable in the context of interest.

Under assumption A.1,

∥

∥

∥

∥

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

∥

∥

∥

∥

≤2σ(M − 1)

M= σ. (3.13)

Assumption A.2 : Given the initialization vector x0, the set Ω0 ≡ x : J(x) <

J(x0) is compact.

In our application, the resource allocation domain, either a frequency band or

a time interval (or both), is always a compact set. The incorporation of the

98


frequency and/or time interval limits in our problem can be done either imposing

box constraints on our optimization or by adding a barrier to the interference

profiles Ii(xi), for all i, i.e., a positive continuous term that starts from the

boundary of the resource domain and goes to infinity linearly, with constant

derivative σ, in order to keep satisfying Assumption A.2 1. Under this choice,

Assumption A.1 holds true. In the following derivations, we follow this second

approach. We are now able to state the main theorems on swarm behavior.

Unbounded Repulsion

Theorem 7 : Let us consider the swarming algorithm described in (3.3), with

attraction/ repulsion functions given by (3.7) and (3.8). Under assumptions A.1

and A.2, the state x(t) converges to the largest invariant subset of the set Ωe =

x ∈ Ω0 : x = 0 and the modulus of the displacement vector is upper and lower

bounded as follows:

β2 ≤ ‖∆‖ ≤ γ2, (3.14)

where

β2 = −σ

2cAλM (L)+

1

2

√

(

σ

cAλM (L)

)2

+ 2cRcA

tr(D)

λM (L)(3.15)

and

γ2 =σ

2cAλ2(L)+

1

2

√

(

σ

cAλ2(L)

)2

+ 2cRcA

tr(D)

λ2(L)(3.16)

with tr(D) denoting the trace of D.

Proof. In the following, we drop the dependency on time t, to avoid an ex-

cessive overcrowding of the formulas, and we introduce the notation ∇Tx :=

1To preserve C1 continuity it is also necessary to ensure a smooth transition from the original

interference profile to the modified profile incorporating the barrier.

99


(∇Tx1, . . . ,∇T

xM). The evolution of the time derivative of the global potential

function (3.2) along the trajectory described by (3.3) is

J(x) = [∇xJ(x)]T x =

M∑

i=1

[∇xiJ(x)]T xi

=

M∑

i=1

[−xi]T xi = −

M∑

i=1

‖xi‖2 ≤ 0 ∀t. (3.17)

This means that, while moving along the trajectory given by (3.3), the potential

function J(x) is always nonincreasing and it stops decreasing (i.e., J(x) = 0)

only if xi = 0, ∀i = 1, . . . ,M . If the set defined as Ω0 ≡ x : J(x) < J(x0) is

compact, then using LaSalle’s Invariance Principle [179], we can conclude that, as

t→∞, the state x(t) converges to the largest invariant subset of the set defined

as

Ω1 ≡ x ∈ Ω0 : J(x) = 0 ≡ x ∈ Ω0 : x = 0. (3.18)

Let us consider now the displacement vector ∆i. The time derivative of the

distance ∆i is given by

∆i = xi − ˙x = −∇xiIi(xi) +1

M

M∑

j=1

∇xjIj(xj) +

M∑

j=1

aij g(xj − xi). (3.19)

Defining a cumulative Lyapunov function as V =∑M

i=1 Vi, with Vi =12‖∆i‖

2,

and taking its time derivative along the system trajectory (3.19), we can write

V =

M∑

i=1

Vi =

M∑

i=1

∆Ti ∆i = −

M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i +

+

M∑

i=1

M∑

j=1

aij [ga(‖xj − xi‖)− gr(‖xj − xi‖)] (xj − xi)T∆i (3.20)

From (3.20), the time derivative of the Lyapunov function along the system tra-

jectory can be rewritten as

V = −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i+

100


+M−1∑

i=1

M∑

j=i+1

aijg(‖xj − xi‖)](xj − xi)T∆i +

+

M−1∑

i=1

M∑

j=i+1

aijg(‖xi − xj‖)(xi − xj)T∆j

= −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i +

−1

2

M∑

i=1

M∑

j=1

aij ga(‖xj − xi‖)‖xj − xi‖2 +

+1

2

M∑

i=1

M∑

j=1

aij gr(‖xj − xi‖)‖xj − xi‖2. (3.21)

Exploiting the features of linear attraction and unbounded repulsion of the cou-

pling function g(·), as expressed in (3.7) and (3.8), we obtain

V = −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i +

−cA2

M∑

i=1

M∑

j=1

aij‖xj − xi‖2 +

cR2tr(D). (3.22)

The state vector x can always be decomposed into the motion of the center of

the swarm plus the displacement vector ∆, as follows:

x = 1M ⊗ x+∆ (3.23)

where the disagreement vector ∆ = (∆T1 , . . . ,∆

TM )T ∈ R

nM satisfies ∆ ⊥ 1M ⊗

el, l = 1, . . . , n, where el is the vector of the canonical basis, with all entries

equal to zero, except the l-th component, equal to one. The vector ∆ belongs

to an (nM − n)-dimensional subspace (the disagreement eigenspace of L) that

is orthogonal to the nullspace of L. As a consequence of (2.82) and (3.23),

expression (3.22) can be recast as

V = −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i − cA∆T L∆+

cR2tr(D) (3.24)

101


Applying (2.80) and considering the expression of λ(L) in (2.83), we obtain an

upper bound

V ≤ −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i − cAλ2(L)‖∆‖2 +

cR2tr(D)

(3.25)

and a lower bound

V ≥ −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i − cAλM (L)‖∆‖2 +cR2tr(D)

(3.26)

of the time derivative of the Lyapunov function. Exploiting the Cauchy-Schwarz

inequality and the assumption A.1, expressions (3.25) and (3.26) can be further

bounded as

V ≤ −cAλ2(L)‖∆‖2 + σ

M∑

i=1

‖∆i‖+cR2tr(D), (3.27)

V ≥ −cAλM (L)‖∆‖2 − σM∑

i=1

‖∆i‖+cR2tr(D) (3.28)

Now, considering the inequality∑M

i=1 ‖∆i‖ ≤√

∑Mi=1 ‖∆i‖2 = ‖∆‖, the final

expressions for the lower and upper bounds of the time derivative of the global

Lyapunov function take the form

V ≤ −cAλ2(L)‖∆‖2 + σ‖∆‖+

cR2tr(D), (3.29)

V ≥ −cAλM (L)‖∆‖2 − σ‖∆‖ +cR2tr(D). (3.30)

Let us consider now the two bounds in more detail.

Upper Bound

For the upper bound case, expression (3.29) assumes the form

V ≤ −c1(

‖∆‖2 − c2‖∆‖ − c3)

, (3.31)

102


with c1, c2, c3 > 0. The RHS of (3.31) is a polynomial in ‖∆‖, having two

real roots, with opposite sign, which we denote, respectively, as γ1 and γ2, with

γ2 > γ1. Clearly, only the positive root γ2 is feasible, as ‖∆‖ is certainly non-

negative. Introducing the 2D plane having axes V and ‖∆‖, the evolution of the

system occurs below the solid parabola shown in Fig. 3.3. Clearly, at convergence,

i.e., when V = 0, it must be

‖∆‖ ≤ γ2. (3.32)

where

γ2 =σ

2cAλ2(L)+

1

2

√

(

σ

cAλ2(L)

)2

+ 2cRcA

tr(D)

λ2(L). (3.33)

This result proves the swarm cohesiveness.

Lower Bound

For the lower bound case, expression (3.30) assumes the form

V ≥ −c4(

‖∆‖2 + c5‖∆‖ − c6)

, (3.34)

with c4, c5, c6 > 0. Also in this case, the RHS of (3.34) is a parabola, with two

real roots having opposite sign, which we indicate as β1 and β2, with β2 > β1.

Only β2 is feasible in this case. In this case, the evolution of the system occurs

in the space above the dashed parabola depicted in Fig. 3.3. At convergence, it

must be

‖∆‖ ≥ β2, (3.35)

where

β2 = −σ

2cAλM (L)+

1

2

√

(

σ

cAλM (L)

)2

+ 2cRcA

tr(D)

λM (L). (3.36)

This proves that the vectors ∆i cannot become all zero, i.e., the vectors xi cannot

collapse all to the same value. This is a result of the repulsion force.

103


||∆||

V.

β2

γ2θ

uβ

1θ

lγ1

Figure 3.3: Upper and lower bounds of the potential function time derivative.

Combining both upper and lower bounds, the evolution of the system occurs in

the dashed area sketched in Fig. 3.3 and then, at convergence, (3.14) must hold

true. We can easily check that, indeed, β2 < γ2. This concludes our proof.

Remark: The inequality (3.14) implies that the modulus of ‖∆‖ cannot be

zero, avoiding the overall collapse on the swarm center, but does not prevent

some pairs of nodes to end up with the same resource allocation vector. This

does not happen if the graph is fully connected, as in such a case the repulsion

between nodes with the same allocation vector would go to infinity. However, in

a sparse graph, two nodes i and j with no direct link between them (i.e., with

aij = 0), may end up with the same resource vector. Actually, in our intended

application, where the coefficients aij depend on the distance between the nodes,

it may happen that two nodes get the same resource vector only if they are

not neighbors. But this is indeed a positive behavior as it gives rise to what is

typically known as spatial reuse of frequency slots. In Section V we will exhibit

some numerical results showing that the proposed approach is intrinsically able

to provide a spatial reuse of frequencies.

104


Bounded Repulsion

Theorem 8 : Let us consider the swarming algorithm described in (3.3), with

attraction/ repulsion functions given by (3.7) and (3.9). Under assumptions A.1

and A.2, the state x(t) converges to the largest invariant subset of the set Ωe =

x ∈ Ω0 : x = 0 and the modulus of the displacement vector is upper bounded

as follows:

‖∆‖ ≤ γ, (3.37)

where

γ =σ + cRdmax

cAλ2(L). (3.38)

Proof. The proof of the system stability follows the same steps as in precedence.

In the following, we derive the upper bound γ on the displacement vector ∆ in the

case of bounded repulsion as in (3.9). From equation (3.20), the time derivative

of the Lyapunov function along the system trajectory can be rewritten as:

V = −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i +

−1

2

M∑

i=1

M∑

j=1

aij ga(‖xj − xi‖)‖xj − xi‖2 +

−M∑

i=1

M∑

j=1

aij gr(‖xj − xi‖)(xj − xi)T∆i. (3.39)

As a consequence of (2.82) and (3.23), expression (3.39) can be recast as

V = −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i − cA∆T L∆+

−M∑

i=1

M∑

j=1


105


Applying (2.80) and considering the expression of λ(L) in (2.83), we obtain the

upper bound

V ≤ −M∑

i=1

[

∇xiIi(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i − cAλ2(L)‖∆‖2 +

−M∑

i=1

M∑

j=1


Now, exploiting the Cauchy-Bunyakovsky-Schwarz (CBS) inequality, assumption

A.1, and the inequality in (3.9), expression (3.41) can be further bounded as:

V ≤ −cAλ2(L)‖∆‖2 + σ

M∑

i=1

‖∆i‖+M∑

i=1

dii‖∆i‖, (3.42)

where dii =∑M

j=1 aij is the connectivity degree of node i. Then, using the

inequality∑M

i=1 ‖∆i‖ ≤ ‖∆‖ and denoting with dmax the maximum connectivity

degree of the network, expression (3.42) can be bounded as:

V ≤ −cAλ2(L)‖∆‖2 + (σ + cRdmax)‖∆‖

≤ ‖∆‖ [σ + cRdmax − cAλ2(L)‖∆‖] . (3.43)

Arguing as in Theorem 1, the evolution of the system in the plane (V , ‖∆‖) occurs

in the region below the parabola described by equation (3.43), which intersects

the ‖∆‖-axis in zero and γ. In particular, at convergence, i.e. when V = 0, we

must have

‖∆‖ ≤ γ, (3.44)

where

γ =σ + cRdmax

cAλ2(L). (3.45)

This result proves the swarm cohesiveness.

Remark: As we have shown before, unbounded repulsion determines the exis-

tence of both an upper bound and a lower bound on the swarm size. On the

106


contrary, in the bounded repulsion case, the swarm admits only an upper bound

on the displacement vector ‖∆‖. A lower bound is not guaranteed as the effect

of a sufficiently strong interference profile could determine the overall collapse of

the swarm into a single point. This occurrence can be avoided by selecting the

repulsion coefficient cR large enough with respect to the profile effect.

Theorems 1 and 2 have been derived under the assumption of a bounded norm

of the profile gradient. In the following, we will provide closed form expressions

for the lower and upper bounds in the case of an unbounded gradient, e.g., in the

case in which the profile is quadratic.

3.4.2 Quadratic Profile

We consider now a quadratic interference profile given by

I(y) =Aσ

2‖y − cσ‖

2 + bσ (3.46)

where Aσ ∈ R, bσ ∈ R, and cσ ∈ Rn. Its gradient at a point y ∈ R

n is

∇yI(y) = Aσ(y − cσ). (3.47)

Unbounded Repulsion

Theorem 9 : Given the swarming algorithm described in (3.3), with attraction/

repulsion functions given by (3.7) and (3.8), under the assumption of a quadratic

profile as in (3.46), we have

β ≤ ‖∆‖ ≤ γ (3.48)

where the lower and upper bounds are respectively given by

β =

√

cRtr(D)

2(cAλM (L) +Aσ), γ =

√

cRtr(D)

2(cAλ2(L) +Aσ). (3.49)

with β ≤ γ.

107


This proves swarm cohesiveness and prevents the individuals to collapse on the

swarm center. Moreover, if the network is fully connected, the swarm converges

to the only stable equilibrium of the system, given by

‖∆‖ =

√

cRM(M − 1)

2(cAM +Aσ). (3.50)

Proof. Under the hypothesis of quadratic profiles we have

[

∇xiI(xi)−1

M

M∑

j=1

∇xjIj(xj)

]T

∆i = Aσ‖∆i‖2. (3.51)

As a consequence, the upper bound (3.25) can be rewritten as

V ≤ −cAλ2(L)M∑

i=1

‖∆i‖2 −Aσ

M∑

i=1

‖∆i‖2 +

cR2tr(D)

= −[

cAλ2(L) +Aσ

]

M∑

i=1

‖∆i‖2 +

cR2tr(D)

= −[

cAλ2(L) +Aσ

][

‖∆‖2 − γ2]

. (3.52)

Similarly, the lower bound (3.26) becomes

V ≥ −cAλM (L)

M∑

i=1

‖∆i‖2 −Aσ

M∑

i=1

‖∆i‖2 +

cR2tr(D)

= −[

cAλM (L) +Aσ

]

M∑

i=1

‖∆i‖2 +

cR2tr(D)

= −[

cAλM (L) +Aσ

] [

‖∆‖2 − β2]

. (3.53)

Arguing as in Theorem 1, the evolution of the system in the plane (V , ‖∆‖) occurs

in the region comprised between the two parabolas described by equations (3.52)

and (3.53), both centered on the axis ‖∆‖ = 0. In particular, at convergence, i.e.

when V = 0, we must have

β ≤ ‖∆‖ ≤ γ, (3.54)

108


where

β =

√

cRtr(D)

2(cAλM (L) +Aσ), γ =

√

cRtr(D)

2(cAλ2(L) +Aσ), (3.55)

with β ≤ γ. If the network is fully connected, expressions (3.52) and (3.53) hold

with strict equality and the swarm converges to the unique stable equilibrium

point of the dynamical system. The final convergence value can be obtained

substituting, in one of the bounds in (3.55), tr(D) = M(M − 1) and λ2(L) =

λM (L) =M , thus achieving

V = 0 ⇒ ‖∆‖ =

√

cRM(M − 1)

2(cAM +Aσ). (3.56)

This concludes the proof of the theorem.

Bounded Repulsion

Theorem 10 : Given the swarming algorithm described in (3.3), with attrac-

tion/ repulsion functions given by (3.7) and (3.9), under the assumption of a

quadratic profile as in (3.46), we have

‖∆‖ ≤ γ (3.57)

where the lower and upper bounds are respectively given by

γ =cRdmax

cAλ2(L) +Aσ. (3.58)

Proof. Exploiting (3.51), the upper bound (3.41) can be written as:

V ≤ −cAλ2(L)M∑

i=1

‖∆i‖2 −Aσ

M∑

i=1

‖∆i‖2 + cR

M∑

i=1

dii‖∆i‖

= −[

cAλ2(L) +Aσ

]

M∑

i=1

‖∆i‖2 + cRdmax

M∑

i=1

‖∆i‖

≤ −[

cAλ2(L) +Aσ

]

‖∆‖[

‖∆‖ − γ]

. (3.59)

109

3.5 Swarming in the Frequency Domain

As in Theorem 2, the evolution of the system in the plane (V , ‖∆‖) occurs in

the region below the parabola described by equation (3.59), which intersects the

‖∆‖-axis in zero and γ. In particular, at convergence, i.e. when V = 0, we must

have

‖∆‖ ≤ γ, (3.60)

where

γ =cRdmax

cAλ2(L) +Aσ. (3.61)

This concludes the proof of the theorem.

A numerical validation of our theoretical findings will be provided in the Sections

2.6 and 2.7. In the next section, we consider in more detail the situation where the

resource domain is monodimensional (1D), like the frequency axis, for example.

In the ensuing section we will then provide some examples of application for 2D

allocation domain, e.g. time-frequency domain.


In this section we focus our attention on the swarming over a 1D domain, for

example the frequency axis. First, we provide a local stability analysis useful to

show that the introduction of the attraction and unbounded repulsion terms in

the functional to be minimized does not affect the stability of the system. More-

over, considering the swarm discrete-time model, we provide an upper bound for

the step size, depending on simple system parameters, that assures the local con-

vergence to an equilibrium point. Then, we propose fast swarming methods based

on a proper selection of the descent direction of a scaled gradient optimization.

The first method is an approximation of the Newton-based optimization and im-

proves the convergence speed of the algorithm; the second method adapts the

swarming speed with respect to the interference power perceived by the swarm.

Finally, we show some numerical results to assess the performance of the resource

allocation technique based on swarming.

110


3.5.1 Local Stability Analysis

In this section we analyze the behavior of the swarm in proximity of a solution

point x∗ where the gradient of the potential function J(x) is equal to zero. To

study the local stability of the system around a solution point, we consider the

second-order Taylor series expansion of the scalar-valued function J(x) around

x = x∗, given by

J(x) ≃ J(x∗) +∇xJ(x∗)T (x− x∗) +

1

2(x− x∗)TH(x∗)(x− x∗), (3.62)

where H(x∗) is the Hessian matrix, computed in x∗, whose entries are:

Hij(x) :=∂2J(x)

∂xj∂xi. (3.63)

If H(x∗) is positive definite, the potential function J(x) is locally approximated

by a positive definite quadratic form, in the neighborhood of x∗. This guarantees

the local stability of the system. In the one-dimensional case, xi ∈ R and x =

(x1, . . . , xM )T ∈ RM . Considering a nonlinear coupling function characterized

by linear attraction and unbounded repulsion as given in (3.7) and (3.8), the

potential function J(x) assumes the form

J(x) =M∑

i=1

Ii(xi) +1

4

M∑

i=1

M∑

j=1

aij[

cA(xj − xi)2 − cR log(xj − xi)

2]

. (3.64)

The entries Hij(x) of the Hessian matrix are then

Hii(x) =∂2J(x)

∂x2i=

∂2Ii(xi)

∂x2i+

M∑

j=1

aij

[

cA +cR

(xj − xi)2

]

,

Hij(x) =∂2J(x)

∂xj∂xi= − aij

[

cA +cR

(xj − xi)2

]

= Hji(x). (3.65)

According to Gershgorin theorem, H(x∗) is a positive definite matrix with all

the eigenvalues greater than zero if

Hii(x∗) >

∑

j

|Hij(x∗)| , ∀i = 1, . . . ,M. (3.66)

111


In our case, this gives

∂2Ii(x∗i )

∂x2i+

M∑

j=1

aij

[

cA +cR

(x∗j − x∗i )

2

]

>M∑

j=1

aij

∣

∣

∣

∣

cA +cR

(x∗j − x∗i )

2

∣

∣

∣

∣

⇒∂2Ii(x

∗i )

∂x2i> 0 , ∀i = 1, . . . ,M. (3.67)

As a consequence, the convexity of the interference profiles Ii(xi), evaluated in the

system equilibrium point x∗, guarantees the local stability. This is an important

result as it shows that the introduction of the attraction and unbounded repulsion

terms in the functional to be minimized does not affect the system stability.

3.5.2 Discrete-Time Implementation

The swarm evolution has been described, up to now, in continuous time. In

practice, the exchange of information between nodes of the networks requires a

discrete-time implementation. The time discretization of (3.3) yields

xi[k + 1] = xi[k] + α

[

−∂Ii(xi[k])

∂xi+

+

M∑

j=1

aij

(

cA −cR

(xj [k]− xi[k])2

)

(

xj[k]− xi[k])

]

, (3.68)

where the step size α must be sufficiently small to ensure convergence. In the

following, we will provide some upper bounds on ǫ, in order to guarantee conver-

gence, at least in the neighborhood of the solution points.

In the neighborhood of the equilibrium point x∗, the continuous-time dynam-

ical system can be approximated as

x = −∇xJ(x) ≃ −H(x∗)(x− x∗). (3.69)

The discretization of this vector differential equation leads to the following dif-

ference equation

x[k + 1] = x[k]− αH(x∗)(x[k]− x∗) = αH(x∗)x∗ + (I − αH(x∗))x[k]

= αH(x∗)x∗ +W (x∗)x[k] (3.70)

112


where W (x∗) = I −αH(x∗) represents the iteration matrix of the discrete time

algorithm evaluated at the equilibrium point. In order to assure the convergence

of the swarming algorithm, the discrete time mapping must be a contraction

having fixed point x∗ such that x∗ = αH(x∗)x∗+W (x∗)x∗. In this way, letting

y = x− x∗, the discrete time procedure can be rewritten as

y[k + 1] =W (x∗)y[k], (3.71)

and the error vector y converges to zero if the spectral radius (W (x∗)) of the

iteration matrix is less than one in modulus. This condition holds true if

H(x∗) ≻ 0 and 0 < α <2

λMAX(H(x∗)). (3.72)

From the analysis carried out in section 3.5.1, the positive definiteness of the

Hessian matrix H(x∗) is assured by the convexity of the interference profiles

evaluated at x∗. Under this condition, the step size ǫ must be chosen in order

to satisfy the previous bound. The upper limit may be difficult to evaluate

in a distributed manner. Nevertheless, using again Gershgorin’s theorem, the

maximum eigenvalue of H(x∗) can be upper bounded as

λMAX(H(x∗)) ≤ maxi

∂2Ii(x∗i )

∂xi2+ 2

∑

j

aij

(

cA +cR

(x∗j − x

∗i )

2

)

< σ′′

MAX + 2

(

cA +cR

r2MIN

)

dMAX = λσM (3.73)

where σ′′

MAX is the maximum convexity of the interference profile, r2MIN =

mini,j ‖x∗j − x

∗i ‖

2 is determined by the repulsion constant cR and dMAX is the

maximum degree of the network connectivity. Hence, the convergence of the

discrete time algorithm is ensured by choosing the step size ǫ in the interval

0 < α <2

λσM, (3.74)

whose upper bound depends now on global parameters that can be exchanged

through the network.

113


3.5.3 Fast Swarming Algorithms

One of the main drawbacks of gradient-based methods is their speed of con-

vergence, which is known to be low. Clearly, a distributed technique is amenable

for resource allocation only if it guarantees convergence in a few iterations. In

this section we modify the basic swarming algorithm (3.3) in order to increase its

convergence speed. In general, the minimization of the functional in (3.2) can be

achieved through a general scaled-gradient method given by

x = −B(x)∇xJ(x) (3.75)

where B(x) is some positive definite matrix representing a scaling along the di-

rection of steepest descent. The introduction of B(x) is useful to increase the

speed of the algorithm or to enforce particular behaviors on the swarm individ-

uals. The evolution of the time derivative of the global potential function (3.2)

along this modified system trajectory (3.75) is given by

J(x) = [∇xJ(x)]T x = −xT [B(x)]−T x ≤ 0 ∀t. (3.76)

where (·)−T means inverse and transpose of a matrix. This means that, mov-

ing along the trajectory given by (3.3), the potential function J(x) is always

nonincreasing and it stops decreasing (i.e., J(x) = 0) only if x = 0. Un-

der assumption A.2, the set Ω0 ≡ x : J(x) < J(x0) is compact, then us-

ing the LaSalle’s Invariance Principle [179] we can conclude that, as t → ∞,

the state x(t) converges to the largest invariant subset of the set defined as

x ∈ Ω0 : J(x) = 0 ≡ x ∈ Ω0 : x = 0.

Typically, B(x) is chosen equal to the inverse of the Hessian matrix H(x) in

order to implement a Newton recursion having an improved convergence speed

with respect to the normal steepest descent. The Newton’s method approximates

at each iteration J(x) by a quadratic function, as in (3.62), and then it moves

towards the minimum of that quadratic function. However, the matrix H(x) is

a full matrix, with elements given by (3.65). This implies that the computation

114


of B(x) requires a centralized mechanism. One suboptimal, but parallelizable,

solution consists in approximating the Hessian matrix by retaining only its diago-

nal entries. This simplifies the inversion of the Hessian matrix and allows parallel

computation. We consider then the trajectory (3.75), where the diagonal scaling

matrix B(x) has entries given by

Bii(x) =

(

∂2J(x)

∂x2i

)−1

=

(

∂2Ii(xi)

∂x2i+

M∑

j=1

aij

[

cA +cR

(xj − xi)2

])−1

. (3.77)

The i-th element is computable at the correspondent node having access to the

second derivative of the profile at the local point xi and to the positions xj of its

neighbors. For general non convex profiles, the value of Bii(x) can be negative.

To ensure the positive definiteness of the matrix B(x) and to preserve the descent

direction of the algorithm, we can add a scaled identity matrix, with the scale

adjusted at each iteration, to the Hessian’s approximation.

An alternative solution to improve the convergence speed of the swarming

algorithm uses a scaling matrix whose diagonal elements are functions of the

power Ii(xi) perceived at each node i. The goal is to accelerate the motion of

the resources perceiving a high interference and, at the same time, to slow down

the resources that are allocating on idle sub-bands. This adaptive feature can be

implemented using a variable step size depending on a monotonically increasing

function f(Ii(xi)) of the perceived interference power. The values of this function

are lower and upper bounded by positive values ensuring that the matrix B(x)

is positive definite. Examples include linear, quadratic, logarithmic functions

etc...The diagonal entries of the scaling matrix can then be expressed as

Bii(x) = f(Ii(xi)), f(·) ∈ [fmin, fmax] > 0. (3.78)

This solution improves the reaction time needed by the algorithm to perform a

resource allocation on idle bands in case of a PU’s activation. The discrete-time

recursion of node i can then be expressed as

xi[k + 1] = xi[k] + αBii(x[k])

[

−∂Ii(xi[k])

∂xi+

115


+M∑

j=1

aij

(

cA −cR

(xj [k]− xi[k])2

)

(

xj[k]− xi[k])

]

, (3.79)

where Bii(x[k]) is given by (3.77) or (3.78). In the following sections, we will

illustrate how these solutions outperform the convergence speed of the gradient-

based algorithm.

3.5.4 Numerical Examples

In this section we provide some numerical results to assess the performance

of the proposed algorithms.

Example 1 - Validation of the theoretical results for profiles with bounded gradi-

ent: In this example we show some numerical results supporting Theorem 1. We

consider the one-dimensional evolution of 10 agents constituting the swarm in

the presence of a bounded profile composed of the superposition of several Gaus-

sian functions. Each agent interacts with its neighbors according to a connected

0.7 0.8 0.9 1 1.1 1.21

2

3

4

5

6

7

8

9

10

Covering radius r0

||∆

||

||∆||

Upper Bound − γ2

Lower Bound − β2

Figure 3.4: Swarm size parameter versus the node covering radius.

116


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

Attraction parameter cA

||∆

||||∆||

Upper Bound − γ2

Lower Bound − β2

Figure 3.5: Swarm size parameter versus the attraction parameter cA.

topology and updates its intended position according to (3.3), implemented in

discrete-time. To provide a validation of our theoretical findings, we report, on

of Fig. 3.4, the behavior of the swarm size parameter ‖∆‖ and its theoretical

bounds in (3.15) and (3.16), versus the node’s covering radius that determines

the network connectivity. The results have been averaged over 100 independent

realizations. The attraction and repulsion parameters used in this simulation are

cA = 0.2 and cR = 0.2. As we can see, the effect of an increment of connectivity

slightly reduces the swarm size and the swarm parameter ‖∆‖ remains always

inside the theoretical bound interval. The increment of the network connectivity

implies tighter bounds that converge on constant values if the network is fully

connected.

A further example is given on Fig. 3.5, where we show the behavior of the

swarm size parameter ‖∆‖ and its theoretical bounds in (3.15) and (3.16), versus

the swarm attraction constant cA. As expected, an increment of the attraction

force, while keeping constant the repulsion, decreases the swarm size. Also in

this case, we can see how the theoretical bounds are always satisfied.

117


0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Figure 3.6: Network topology and allocation example.

Example 2 - Collision avoidance and spatial reuse of frequency: One of the main

features of the proposed swarming technique is the capability to prevent collisions

between nearby nodes while, at the same time, allowing for spatial reuse of the

frequency channels from secondary nodes far away from each other. The coupling

coefficients aij in (3.3) depend on the distance between the SUs and reflect the

network topology, as dictated by the coverage radius of each node. Interestingly,

as will be shown next, the proposed swarming algorithm leads naturally to spatial

reuse of frequencies, as the network topology becomes more and more sparse. To

quantify the spatial reuse of frequencies, we introduce the reuse factor ζ, defined

as the ratio between the number of resources (frequency subchannels) necessary

to guarantee one slot for each node and the number of channels really allocated

by the algorithm. As an example of channel allocation, in Fig. 3.6 we consider a

network composed of 100 nodes, where each node senses the interference spectrum

shown in Fig. 3.7. The swarm parameters are cA = 0.01 and cR = 0.1. Every

node starts from a random initial position on the spectrum, and then it updates

118


15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20−0.5

0

0.5

1

1.5

Frequency (MHz)

PS

D (

mW

/Hz)

Figure 3.7: Interference profile and allocation example.

its intended position according to (3.3), implemented in discrete-time. In the

application at hand, there is an intrinsic quantization of the frequency resources

given by the subchannel bandwidth. In our implementation, we let the system

evolve according to (3.68) until successive differences in allocation become smaller

than the bandwidth of a frequency subchannel. At that point, the evolution stops

and every SU is allowed to transmit over the selected channel. The final choice is

indicated by assigning a different shape to different subchannels, as shown in the

topology plot reported in Fig. 3.6. In our experiment, the number of available

channels with low interference is eight, hence much smaller than the number

of users. Interestingly, from Fig. 3.6 we can observe that the nodes that have

picked up the same channel are never neighbor of each other. This is indeed one

of the most interesting features of the proposed algorithm. This means that the

algorithm is capable of implementing a decentralized mechanism for spatial reuse

of frequencies. In this case, the network frequency reuse parameter is ζ = 12.5.

Clearly, the reuse factor depends on the sparsity of the graph describing the

network topology. To quantify the effect of the coverage radius of each node

on the reuse factor, in Fig. 3.8 we report the average behavior of ζ, averaged

over 100 independent initializations, versus the covering radius of each node,

119


0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

Covering radius r0

Fre

quency r

euse p

ara

mete

r ζ

CR

= 0.1

CR

= 0.2

CR

= 0.3

Figure 3.8: Frequency reuse parameter versus covering radius.

considering three different values of the repulsion constant cR. In this simulation

we consider M = 15 nodes, cA = 0.2, the same interference profile as in the

bottom side Fig. 3.6 and 15 available channels on the idle band in the middle

of the spectrum. As expected, the behavior of ζ is monotonically decreasing and

it reaches the unit value when the covering radius is such that the network is

fully connected. Furthermore, Fig. 3.8 shows also that, by reducing the value of

cR, the repulsion force is weaker and this facilitates the reuse of frequency slots.

Hence, the parameter cR has to be chosen as a trade-off between reuse factor

and number of collisions. We have checked numerically that in all simulations,

choosing appropriately the swarm parameters, the final channel allocation never

determines collisions among spatial neighbors.

Example 3 - Convergence speed: One of the main issues for distributed resource

allocation algorithms is convergence speed. In this section we show some nu-

merical examples to evaluate the convergence time of the proposed allocation

algorithm and its modified version given in (3.79). The first example assumes the

same settings as in Fig. 3.6. To assess convergence time, in Fig. 3.9, we report

120


0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

Iteration Index

Norm

aliz

ed S

yste

m P

ote

ntial F

unction

Fully connected

r0 = 0.8

r0 = 0.5

Figure 3.9: Normalized system potential function vs. time index, for different

coverage radii.

the average behavior of the evolution of the system potential function (3.9), nor-

malized with respect to the maximum and the minimum value, averaged over 500

independent realizations, vs. the iteration index. Three different coverage radii

are considered to evaluate the impact of network topology on the convergence

speed. The attraction and repulsion parameters of the swarm for this simulation

are cA = 0.2 and cR = 0.2, and the step size is equal to α = 0.05. As expected,

the convergence rate increases as the connectivity increases.

The second example compares the convergence speed of the gradient based

swarming algorithm in (3.3) and of the approximated Newton method with scal-

ing coefficients given by (3.77). In Fig. 3.10, we report the average behavior of

the evolution of the system potential function, e.g., (3.2), normalized with respect

to the maximum and the minimum value, averaged over 500 independent realiza-

tions, vs. the iteration index. The evolution of the gradient-based algorithm is

given by the dashed curve while the approximated Newton version is depicted by

the continuous curve. In this simulation we consider the same interference profile

and network topology depicted in Fig. 3.6. The parameters are the same of the

121


0 5 10 15 20 25 30 35 40 45 50

0

0.2

0.4

0.6

0.8

1

iteration index

Norm

aliz

ed S

yste

m P

ote

ntial F

unction

Gradient Descent

Newton Approx

Figure 3.10: Normalized system potential function vs. time index, for differ-

ent descent directions of the algorithm : gradient descent (dashed) and Newton

approximation (solid).

previous simulation and the step size for the approximated Newton version is

equal to α = 0.8. The step sizes were empirically decided because slightly greater

values determine instability of the algorithm in both cases. From Fig. 3.10, we

can notice how the approximated Newton-scaled version greatly outperforms the

gradient based algorithm. This means that the convergence time of the swarm-

ing algorithm can be considerably improved if every node is able to evaluate the

second order derivative of the system potential function given by (3.79).

Example 4 - Dynamic response of the swarm to a predator (interferer): Natural

swarms are adaptive systems whose individuals cooperate in order to improve

their food search capabilities and to increase their robustness against preda-

tors’ attacks. We show next that the proposed resource allocation increases,

as a by-product, the network robustness against the intrusion of a primary user

(predator). We consider again the network topology depicted in Fig. 3.6, plus

the inclusion of two PU’s that start emitting, at different times, thus causing a

dynamic change of the occupied spectrum. Our goal is to test the dynamic

122


0 50 100 150 2000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Iteration index

Ave

rag

e in

terf

ere

nce

leve

l

Figure 3.11: Dynamic resource allocation by swarming: Reaction time to PU’s

activations, for basic swarming.

response of the network to this changing environment. Playing again with the

swarm analogy, PU’s take now the role of predators whose positions must be

avoided by the swarm individuals.

To give an example of the reaction time needed by the algorithm to react

to the PU’s intrusion and adjust the resource allocation consequently, in Fig.

3.11 we show the behavior of the average interference perceived by the swarm

versus the time index. The two peaks at the iterations 67 and 123 correspond to

the two PU’s activation times. The low power value represents the noise level.

In particular, we compare the results of the gradient-based algorithm in (3.3),

in Fig. 3.11, and its scaled version in (3.78), in Fig. 3.12, that adapts the

convergence speed with respect to the perceived interference. The attraction and

repulsion parameters used in this simulation are cA = 0.2 and cR = 0.2; the step

size is equal to α = 0.05 for both algorithms. We can notice how the adaptive

scaled version needs only a small number of iterations to leave the PU’s regions,

thus outperforming the gradient-based version. This positive behavior is given

by the adaptation of the algorithm with respect to the perceived interference,

123


0 50 100 150 2000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4A

vera

ge in

terf

eren

ce le

vel

Iteration index

Figure 3.12: Dynamic resource allocation by swarming: Reaction time to PU’s

activations, for adaptive scaling.

determining that resources allocating on high interference regions move faster

due to the increment of the profile gradient and the cohesion force.

These examples show that the cohesion force represents an intrinsic robust-

ness factor of the algorithm. In fact, resources allocating over high interference

bands might measure a flat spectrum, thus resulting in limited capabilities to

move out of (flat) occupied bands, if the only cause of change is spectrum gra-

dient. However, increasing the cohesion force, the agents allocating over the low

interference band tend to form cohesive blocks that exert an attraction towards

the agents trapped by mistake over the flat regions of the spectrum occupied by

the primary users. This is an example of cooperation gain.

Example 5 - Comparison with deterministic graph coloring methods: In the previ-

ous examples, we have illustrated the capability of the algorithm of implementing

a decentralized mechanism for spatial reuse of frequencies. The swarming algo-

rithm in (3.3) is indeed based on local exchange of data among SU’s, whose

resources select available channels avoiding conflicts only with spatial neighbors.

The aim of this example is to compare the number of iterations needed by the

124


10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

Number of nodes

Ave

rag

e n

um

be

r o

f ite

ratio

ns

Graph Coloring α = 1

Graph Coloring α = 1.25

Graph Coloring α = 1.5

Swarming α = 1

Swarming α = 1.25

Swarming α = 1.5

Figure 3.13: Average number of iterations to obtain convergence versus number

of nodes, for different degrees of network connectivity.

swarming algorithm to convergence with respect to a graph coloring algorithm

[87]– [90]. As a comparison, we consider a deterministic distributed graph color-

ing method from [87]– [88], having the same complexity of the swarming algorithm

in terms of number of exchanged messages per iteration. Several other methods

having a better convergence rate, which is paid by a greater complexity of the

algorithm, are also present in the literature.

In this example, we generate the interference graph among SU’s as a random

geometric graph over a unitary area. The covering radius of each node is chosen

as

r0 = α

√

2 logM

πM(3.80)

which guarantees network connectivity with probability one as the number of

nodes goes to infinity [91]. The selection of the parameter α allows a tuning

of the node’s covering radius, thus affecting the network connectivity. For each

network configuration, we consider the presence of at least dmax+1 available chan-

125

3.6 Swarming in the Time-Frequency Domain

nels, where dmax is maximum number of neighbors of a node (maximum degree)

in the interference graph. In Fig. 3.13, it is illustrated the average behavior of the

number of iterations needed by two different methods to converge versus number

of nodes composing the network, for different degrees of network connectivity.

The behavior of the swarming algorithm is depicted with dashed lines, whereas

the graph coloring method is plotted using solid lines. As we can notice from

Fig. 3.13, the average number of iterations needed by the graph coloring method

to converge grows increasing the number M of SU’s and the connectivity of the

interference graph (tuned by the parameter α). Interestingly, we notice how, fix-

ing the number of network nodes, the swarming algorithm generally outperforms

the graph coloring method, especially for large and highly connected networks.

Indeed, increasing the connectivity of the interference graph, the graph coloring

performance gets worse whereas the swarming algorithm improves its conver-

gence speed. This positive behavior is due to the attraction force of the swarm,

which actually improves the convergence speed of the algorithm, thus helping the

resource allocation performance.


The proposed swarming procedure can be extended to the two-dimensional

domain, representing, for instance, the time-frequency plane. In this case, the vec-

tor xi referred to node i has two entries representing the position of the frequency

subchannel and the time slot that node i intends to occupy. Before occupying

the slot, each secondary node interacts with its neighbors exchanging information

on the intended slot. Every node then updates its intended position according

to (3.3), implemented in discrete-time. In the application at hand, there is an

intrinsic quantization of the time-frequency resources given by the subchannel

bandwidth and by the duration of the time slot. In our implementation, we let

the system evolve according to (3.3) until successive differences in allocation be-

come smaller than the discretization step in the time-frequency domain. At that

126


0 5 10 15 20 25 30 350

5

10

15

20

25

30

Initial positions

Evolution

Final positions

center of the swarm

Figure 3.14: Example of 2D allocation, considering a quadratic profile.

point, the evolution stops and every SU is allowed to transmit over the selected

resource, i.e. a pair of frequency subchannel/time slot. A major difference with

respect to the frequency domain is that the allocation over successive time slots

requires the knowledge of the primary users’ activities through time. Of course,

this noncausal knowledge of the future is not available. However, if we have a

statistical model of the interference activity, we may derive a resource alloca-

tion mechanism, based on our statistical model. In this section we provide some

examples of swarming in the time-frequency plane. We consider first the ideal

case of known interference profile, which can represent a limit case of a static

environment. Then, we will consider the more realistic case in which the PU’s

activity is modeled as a Markov chain.

3.6.1 Swarming in a Static Interference Environment

In the first example we provide numerical support to Theorem 3. In Fig.

3.14 we show the two-dimensional evolution of 15 agents constituting the swarm,

considering the presence of a quadratic profile having a global minimum (Aσ > 0)

127


0 2 4 6 8 10

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

Aσ

||∆

||||∆||

Upper Bound − γ

Lower Bound − β

Figure 3.15: Swarm size versus the magnitude of the quadratic profile Aσ.

at the center of the plane. Each agent interacts with its neighbors according

to the depicted topology and updates its intended position according to (3.3),

implemented in discrete-time. The initial guesses of the agents are represented

as blue dots, scattered randomly across the time-frequency plane. The evolution

is depicted by the red curves and the final allocations are given by the violet dots.

The profile magnitude Aσ is equal to 1 and the attraction/repulsion constants are

cA = 1 and cR = 5. It is evident how the swarm moves toward the minimum of

the profile avoiding collisions among nearby agents. At convergence, the swarm

exhibits an equilibrium configuration characterized by a norm of the distance

vector ‖∆‖ = 7.67, which falls inside the interval constituted by the theoretical

lower and upper bounds, respectively β = 4.45 and γ = 9.22. To verify our

theoretical findings, on Fig. 3.15, we report the behavior of the swarm size

parameter ‖∆‖ and its theoretical bounds in (3.55), versus the magnitude Aσ

of the quadratic profile. The results have been averaged over 100 independent

realizations. The swarm parameters are cA = 1 and cR = 5. As we can see, the

effect of an increment of the profile magnitude reduces the swarm size and the

swarm parameter ‖∆‖ remains always inside the theoretical bound interval. The

128


0 2 4 6 8 10

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

Aσ

||∆

||

||∆||

Upper Bound − γ

Lower Bound − β

Figure 3.16: Swarm size versus the magnitude of the quadratic profile Aσ.

increment of the network connectivity implies tighter bounds on the final swarm

size and the presence of a unique equilibrium point in case of a fully connected

network. This behavior is shown in Fig. 3.16, where we repeat the previous

simulation increasing the coverage radius of each node and, as a consequence, the

connectivity of the network graph. A second example is reported in Fig. 3.17.

The boxes in Fig. 3.17 represent the power allocation of primary users, in the

time-frequency domain, supposed to be static and known, within the convergence

time. These are the regions that do not have to be occupied by the SUs. The

initial guesses of the SUs are represented as squares, scattered randomly across

the time-frequency plane. The evolution of the resource allocation is depicted

by the dotted curves and the final allocations are given by dots. It is evident

how the SUs avoid the positions occupied by the primary users, tend to keep the

spread in the time-frequency plane as small as possible and, at the same time,

they avoid collisions with each other. The previous examples refer to the idealistic

case where the interference is assumed to be static and known, at least within

the convergence time of the swarming algorithm. Next section generalizes the

approach to the case where the interference is known only in a statistical sense.

129


−5 0 5 10 15 20 25 30 35

0

5

10

15

20

25

30

Frequency Axis

Tim

e A

xis

Figure 3.17: Example of time-frequency allocation.

3.6.2 Swarming in the Presence of Markovian Interference

In this section, we model the interference activity, over each frequency sub-

channel, as a two-state continuous-time homogeneous Markovian chain. The

two (on/off) states refer to the cases where the interferer is transmitting or is

idle. This model gives a map of the expected interference power over the time-

frequency plane, conditioned to the measurement performed in the initial time

slot. This expected profile is then used by the swarm algorithm to allocate re-

sources according to the assumed statistical model.

Let us denote by

Qk =

(

−λk µk

λk −µk

)

(3.81)

the transition matrix of the interferer over the k-th frequency subchannel. At each

time t, each subchannel can be either idle or active. Let us denote by Pk0(t) and

Pk1(t) the probabilities that subchannel k, at time t, is idle or active, respectively.

Let us also introduce the probability vector Pk(t) := [Pk0(t),Pk1(t)]T . Given the

130


vector Pk(0) at time 0, the vector probability Pk(t) at time t is

Pk(t) = eQk tPk(0), (3.82)

where

eQk t =

(

1− λkλk+µk

(

1− e−(λk+µk)t) µk

λk+µk

(

1− e−(λk+µk)t)

λkλk+µk

(


1− µkλk+µk

(


)

. (3.83)

Let us suppose that, on each channel, it is known, through preliminary estimation,

the average power pk and the transition rates from idle to idle λk and from active

to active µk. Let us denote by pk(t; 0) the expected power on channel k, at time

t, conditioned to the knowledge of the channel status at time 0.

Suppose now that at time 0, the channel k is sensed as idle. The probability

vector that, at time t > 0, the channel will be either idle or busy is then

Pk(t) =

(

1− λkλk+µk

(


λkλk+µk

(


)

. (3.84)

Conversely, if, at time 0, the channel is sensed as active, the probability vector

at time t is

Pk(t) =

(

µkλk+µk

(


1− µkλk+µk

(


)

. (3.85)

Hence, the expected power, at time t on channel k can be expressed as

pk(t; idle at 0) =λk pkλk + µk

(


, (3.86)

pk(t; active at 0) = pk −µkpkλk + µk

(


. (3.87)

The expressions (3.86) and (3.87) model the time evolution of the expected in-

terference power over each subchannel. This model allows the construction of

a map of the expected interference distribution, conditioned to the estimation

performed in the first slot. This profile is then used by the swarm to dynamically

allocate power over the time-frequency plane. As an example, Fig. 3.18 shows

the evolution of the swarm in the time-frequency plane. The grey level represents

131

3.7 Discrete-Time Distributed Optimization

Frequency Axis

Tim

e A

xis

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

1

2

3

4

5

6

7

8

9

Figure 3.18: Example of time-frequency allocation with Markovian interference.

the expected interference profile. At time 0, there are two disjoint sub-bands with

a high level of interference and three interference-free regions. The evolution of

the (expected) interference in time is given by (3.86), in case the channel at time

0 is sensed as idle, or by (3.87), in case the channel is sensed as busy. The initial

guesses of the SU’s are represented as squares, scattered randomly across the

time-frequency plane. The evolution of the resource allocation is depicted by the

dotted curves and the final allocations are given by dots. This figure shows that

the swarm tends to keep the spread in the time-frequency plane as limited as

possible while avoiding, at the same time, collisions. Interestingly, the swarm

tends to stay as close as possible, to time 0, where the prediction is better, under

the constraint of avoiding collisions.


The swarm evolution (3.3) has been described, so far, mainly in continuous

time. In practice, the implementation is generally performed in discrete-time.

Our goal in this work is the distributed minimization of (3.2). In general, the

132


objective function in (3.2) is not convex in the resource allocation vector x and,

as a consequence, the problem may have multiple local optima. A local solution

can be found in a centralized manner using standard optimization algorithms.

However, we focus on distributed solutions where it is allowed a local coordination

among SUs through a limited exchange of data. A possible way to achieve the

solution in decentralized form is to use a simple gradient based optimization, so

that every node starts with an initial guess, let us say xi[0], and then it updates

its own resource allocation vector xi[k] in time according to the following discrete-

time recursion:

xi[k + 1] = xi[k]− αi[k]∇xiJ(x) (3.88)

= xi[k]− αi[k]

∇xiIi(xi[k])−M∑

j=1

aij g(xj[k]− xi[k])

,

i = 1, . . . ,M , where k is the time index, and αi[k] is a positive iteration-dependent

step-size. In this section, we consider a coupling function g(·) with linear attrac-

tion (3.7) and bounded repulsion (3.10). Under these conditions on the coupling

function g(·), the potential function (3.2) can be written as:

J(x) =

M∑

i=1

Ii(xi) +cA2

M∑

i=1

M∑

j=1

aij‖xj − xi‖2 +

+ cGcR2

M∑

i=1

M∑

j=1

aij exp

(

−‖xj − xi‖

2

cG

)

=M∑

i=1

Ii(xi) +cA2

M∑

i=1

M∑

j=1

aij‖xj − xi‖2

+ cGcR2

M∑

i=1

M∑

j=1

aijexp

(

−‖xj−xi‖2

cG

)

‖xj − xi‖2‖xj − xi‖

2

= tr(Σ(x)) + cAxT (L⊗ In)x+ cGcRx

T (Lr,x ⊗ In)x (3.89)

133


where tr(·) denotes the trace of a matrix. The matrices Σ(x) and Lr,x write as

Σ(x) = diag(I1(x1), . . . , IM (xM )) (3.90)

Lr,x = Dr,x −Ar,x (3.91)

where Ar,x denotes a symmetric state dependent adjacency matrix whose entries

are given by

[

Ar,x

]

ij=

aij‖xj−xi‖2

exp

(

−‖xj−xi‖2

cG

)

,

[

Ar,x

]

ii= 0

(3.92)

In our setup, the foraging profiles Ii(xi) represent the interference power dis-

tribution on the resource domain, hence, the term tr(Σ(x)) is always positive.

Moreover, the matrix Lr,x is positive semi-definite for all x implying that the

corresponding quadratic form in (3.89) is always grater than or equal to zero,

thus leading to J(x) ≥ 0 for all x. Furthermore, considering the choice of a

coupling function g(·) with linear attraction (3.7) and bounded repulsion (3.10)

and resorting to the assumptions A.1-A.2, the potential function J(x) is also

continuously differentiable, with continuous second-order partial derivatives. We

are now able to state the convergence result on the discrete time procedure (3.88).

Theorem 11 Consider the discrete-time swarming algorithm in (3.88) with ar-

bitrary initial state x[0]. Under the assumptions A.1-A.2 and considering a

coupling function g(·) with linear attraction (3.7) and bounded repulsion (3.10),

the algorithm converges to a stationary point of the function J(x) in (3.89), as

k →∞, if the step size α is chosen such that

0 < α <2

L, (3.93)

where L is the Lipschitz constant of the potential function J(x) in (3.89).

Proof. As shown before, the potential function J(x) in (3.89) is continuously

differentiable (Lipschitz-continuous) and bounded from below. Since we are min-

imizing this function using a descent algorithm, Theorem 1 applies and the con-

vergence result follows.

134


3.7.1 Projected Swarming Algorithms

Up to now, we have considered an unconstrained distributed optimization

of the global potential function (3.2). In our derivations, we assumed the opti-

mization set to be compact. In our application, the resource allocation domain,

either a frequency band or a time interval (or both), is always a compact set.

The incorporation of the frequency and/or time interval limits in our problem

can be done either imposing box constraints on our optimization or by adding

a barrier to the interference profiles I(xi), for all i, i.e., a positive continuous

term that starts from the boundary of the resource domain and goes to infinity

linearly, with constant derivative σ, in order to keep satisfying Assumptions A.1

and A.2. In this section, we consider the first approach, imposing box constraints

on the values that the resource allocation vector can assume.

Let us consider the following constrained optimization problem:

minx

J(x)

s.t. xil 4 xi 4 x

iu i = 1, . . . ,M

(3.94)

where x = (xT1 , . . . ,x

TM )T , and 4 denotes component-wise inequality, e.g., a 4 b

if ai ≤ bi, ∀i = 1, . . . , n. The box constraints in (3.94) denote a frequency band

or a time interval (or both) where the i-th cognitive SU can allocate its resources.

As before, the objective function in (3.2) is not concave in the resource allocation

vector x and, as a consequence, the problem may have multiple local optima.

A local optimum x∗ = [x∗1T , . . . ,x∗

MT ]T of problem (3.2) is a regular point 2

and, as a consequence, it satisfies the Karush-Kuhn-Tucker (KKT) conditions

[188]. In particular, we focus on distributed solutions where it is allowed a local

coordination among SU’s through a limited exchange of data. The problem is

amenable for distributed solutions because the optimization set X =∏M

i=1Xi is

given by Cartesian product of sets Xi, allowing the parallel computation of the

algorithm. Then, a possible way to achieve the solution in decentralized form

2A feasible point is said to be regular if the equality constraints gradients and the active

inequality gradients are linearly independent [188].

135


is to use a gradient projection optimization, so that every node starts with an

initial guess, let us say xi(0), and then it updates its own resource allocation

vector xi(k) in time according to the following discrete-time dynamical system:

xi[k + 1] = [xi[k] + α∇xiJ(x[k])]Xi= T i(x[k]), (3.95)

k ≥ 0, i = 1, . . . ,M , where [·]Xi denotes the projection over the feasible set Xi, α

is the step size, and ∇xiJ(x[k]) ∈ Rn is the discrete-time version of (3.3). We are

now able to state the convergence result on the discrete time procedure (3.95).

Theorem 12 Consider the discrete-time swarming algorithm in (3.95) with ar-

bitrary initial state x[0] and let x[k] be the sequence generated by it. Further-

more, consider the assumptions A.1-A.2 and a coupling function g(·) with linear

attraction (3.7) and bounded repulsion (3.10). Then, by selecting the step-size as

0 < α <2

L, (3.96)

where L is the Lipschitz constant of the potential function J(x) in (3.89), if x∗

is an accumulation point of the sequence x[k], the optimal local solution x∗ is

a fixed point of the mapping T (x) = colT i(x)Mi=1, such that x∗ = T (x∗).

Proof. As shown before, the potential function J(x) in (3.89) is continuously

differentiable (Lipschitz-continuous) and bounded from below. Since we are min-

imizing this function using a projection-based descent algorithm, Theorem 2 ap-

plies and the convergence result follows.

In this section, we showed the convergence properties of discrete-time swarming

algorithms, minimizing the global cost function (3.2) in a distributed fashion

using descent approaches. In the next section, we will consider the effect that

realistic channels have on the proposed resource allocation technique based on

swarming, showing how to handle the randomness introduced by fading and noise

in order to ensure the convergence of the proposed techniques.

136

3.8 The Effect of Noise and Realistic Channels


The swarming mechanism studied up to now assumed ideal communications

among the cognitive nodes. However, in a realistic scenario, the wireless channel

is affected by random fading and additive noise, which induce errors in the re-

ceived packets. In such a case, the receiving node could request the retransmission

of the erroneous packets, but this would imply random delays in the communica-

tion among the cognitive nodes and it would be complicated to implement over

a totally decentralized system. It is then of interest to analyze networks where

the erroneous packets are simply dropped. Moreover, the data exchanged among

the nodes is usually quantized using a finite number of bits, and then the effect

of quantization noise on the swarm mechanism should be properly taken into

account. The goal of this section is to extend the swarm-based resource assign-

ment mechanism to the more realistic case where the packets exchanged among

the cognitive nodes, while running the swarm mechanism, are randomly dropped

and the transmitted data are encoded with a finite number of bits.

Several swarm models have been analyzed in the control literature, see, e.g.

[120]- [125], in the case where the graph describing the interaction among the

swarm individuals varies with time, thus inducing a switching topology. In [122]

the authors showed that, if the network graph is always connected, a stable flock-

ing motion can be achieved by using a set of switching control laws given by a

combination of attractive/repulsive and alignment forces. Reference [123] consid-

ered a swarm model affected by a switching topology and proved the convergence

of the swarm to a common velocity vector and the stabilization of inter-agent

distances, regardless of switching, as long as the network remains connected all

the time. The cohesiveness of a hybrid swarm model, suitable to describe swarm

aggregation with limited sensing ability, was also analyzed in [124] taking into

account switching topologies. The flocking behavior of multi-agent systems with

switching topology in a noisy environments was considered in [125], where it was

shown that, although the information is contaminated by noise, all agents can

137


form and maintain the flocking behavior if the gradient of the environment is

bounded and the interaction graph is jointly connected.

The effect of random graphs on consensus algorithms has been thoroughly

studied in a series of works, such as [137]- [141], which focused on the conver-

gence of consensus protocols in the presence of random disturbances. In [137],

the authors use a decreasing sequence of weights to prove the convergence of

consensus protocols to an agreement space in the presence of additive noise un-

der a fixed network topology. A distributed consensus algorithm in which the

nodes utilize probabilistically quantized information to communicate with each

other was proposed in [138]. As a result, the expected value of the consensus

is equal to the average of the original sensor data. A stochastic approximation

approach was followed in [139], which considered a stochastic consensus problem

in a strongly connected directed graph where each agent has noisy measurements

of its neighboring states. The study of a consensus protocol that is affected by

both additive channel noise and a random topology was considered in [141]. The

resulting algorithm relates to controlled Markov processes and the convergence

analysis relies on stochastic approximation techniques. In this work, inspired by

these recent results, we propose a decentralized algorithm to solve the resource as-

signment problem in the presence or random disturbances (such as fading, noise,

and quantization) and we prove its convergence in the presence of random link

failures and quantization noise.

The section is organized as follows. First, we briefly recall some basic concepts

from random link failures model and dithered quantization that will be used

throughout the section. Then, we will demonstrate the stochastic convergence

of the swarming mechanisms previously introduced. We also provide simulation

examples corroborating the theoretical results and showing the effect of the radio

channel impairments on the performance.

138


3.8.1 Random Link Failures

In a realistic communication scenario, some packets may be lost at random

times. To account for this fact, we allow the links among the network nodes to

fail at some probability, inducing a time-varying, or switched, network topology,

depending on the link failures. In this case, we model the network at time k as

an undirected graph, G[k] = V,E[k] where the graph Laplacians is taken as a

sequence of i.i.d. matrices L[k] of the form:

L[k] = L+ L[k] (3.97)

where L denote the mean matrix and L[k] are i.i.d. perturbations around the

mean. We do not make any assumptions on the link failure model. Although

the link failures and the Laplacians are independent over time, during the same

iteration, the link failures can still be spatially correlated (even during the same

iteration). Moreover, connectedness of the graph is an important issue. We do

not require the random instantiations G[k] of the graph be connected for all k; We

only require that the graph is connected on average. This condition is captured by

having the second eigenvalue of the expected Laplacian matrix strictly positive,

i.e., λ2(L) > 0.

3.8.2 Dithered Quantization

We assume that each inter-node communication channel uses a uniform quan-

tizer, which is defined by the following vector mapping, q(·) : Rn → Qn,

q(y) = [b1∆, . . . , bn∆]T = y + e(y), (3.98)

where the entries of the vector y, the quantization step ∆ > 0, and the error e

satisfy

(bm − 1/2)∆ ≤ ym ≤ (bm + 1/2)∆, 1 ≤ m ≤ n,

−∆/2 1n ≤ e(y) ≤ ∆/2 1n, for all y.(3.99)

139


The quantization alphabet is

Qn = [b1∆, . . . , bn∆]T |bm ∈ Z,∀m. (3.100)

Conditioned on the input, the quantization error e(y) is deterministic. This

strong correlation of the error with the input influences the statistical properties

of the error and can influence the convergence of the algorithm. To avoid these

effects, we consider dithered quantization [175]- [176], which endows the quanti-

zation error with some useful statistical properties. The dither added to random-

ize the quantization effects satisfies a special condition, namely the Schuchman

conditions, as in subtractively dithered systems, see [177]. Then, at every time

instant k, adding to each component ym[k] a dither sequence νm[k]k≥0 of i.i.d.

uniformly distributed random variables on [−∆/2,∆/2) independent of the input

sequence, the resultant error sequence ǫm[k]k≥0 becomes

ǫm[k] = q(ym[k] + νm[k])− (ym[k] + νm[k]). (3.101)

The sequence ǫm[k]k≥0 is now an i.i.d. sequence of uniformly distributed ran-

dom variables on [−∆/2,∆/2), which is independent of the input sequence.

Thus, by randomizing the input to the uniform quantizer, we can render the

quantization error to be independent of the input and uniformly distributed on

[−∆/2,∆/2).

3.8.3 Stochastic Convergence

In this section, using stochastic approximation arguments, we demonstrate

the convergence properties of the proposed swarming algorithms impaired by

random link failures, quantization and noise.

Basic Swarming Algorithm

In this section we reformulate the swarming problem as the search for the zeros

of a deterministic function, whose value is corrupted by random disturbance and

140


can be observed at each time instant. We will provide conditions for the almost

sure convergence of the search procedure. For our subsequent derivations, we

consider the assumptions A.1-A.2, and the following condition.

Assumption A.3 : To ensure the swarm cohesion and, hence, a finite swarm

size, we assume that the graph describing the network topology is connected. As

we will see, due to the random nature of the network graph, we only require a

topology connected on average. This condition is captured by having the second

eigenvalue of the expected Laplacian strictly positive, i.e. λ2(L) > 0.

In an ideal case communication case, since the function (3.2) is Lipschitz continu-

ous and bounded from below, it is possible to prove the convergence of a discrete-

time gradient algorithm by using a classical descent approach [189]. Then, by

selecting the step-size of the algorithm sufficiently small (smaller than the inverse

of the Lipshitz constant), the discrete-time version of the iterative procedure (3.3)

will asymptotically converge to a local minimum of the potential function (3.2).

However, in an imperfect communication scenario, where the network links may

fail randomly and communication is corrupted by quantization noise, the nodes

will have access to a random subset of their neighbors and the received data will

likely be corrupted. Furthermore, each node needs to estimate the interference

profile Ii(xi) or at least its gradient for use in (3.3). Estimating this gradient vec-

tor will be subject to errors and we therefore denote the estimate of the gradient

by:

∇xi Ii(xi) = ∇xiIi(xi) + ηi (3.102)

where ηi is a zero mean i.i.d. vector noise sequence of bounded variance. Under

these non-ideal conditions, the convergence of (3.3) to a local minimum is not

assured and the swarming algorithm needs to be adjusted in order to handle the

data imperfections.

The swarm evolution (3.3) has been mainly described, so far, in continuous

time. In practice, the implementation is generally performed in discrete-time.

141


A discrete time version of (3.3) that accounts for random link failures, dithered

quantization noise and estimation errors, can be written as:

xi[k + 1] = xi[k] + α[k]

[

−∇xi[k]Ii(xi[k]) − ηi[k] + (3.103)

+

M∑

j=1

aij [k] g(q(xj [k] + νij [k])− xi[k])

]

, i = 1, . . . ,M,

where α[k] is a positive iteration dependent step-size. Now, exploiting the feature

of subtractively dithered systems in (3.101), the previous expression is given by:

xi[k + 1] = xi[k] + α[k]

[

−∇xi[k]Ii(xi[k])− ηi[k] + (3.104)

+

M∑

j=1

aij[k] g(xj[k]− xi[k] + νij[k] + ǫij [k])

]

, i = 1, . . . ,M.

Starting from some initial position in the resource domain, xi[0] ∈ Rn, each node

generates via (3.104) a sequence of resource allocations, xi[k]k≥0. The position

xi[k + 1] at the i-th node at time k + 1 is a function of: its previous position;

the communicated quantized postions at time k of its neighboring sensors; and

the new estimate of the profile gradient ∇xi[k]I(xi[k]). As described in Section

II-B, the data is subtractively dithered quantized, such that the quantized data

received by the i-th sensor from the j-th sensor at time k is q(xj[k] + νij [k]).

It then follows from the discussion in Section II-B that the quantization error

ǫij [k] is a random vector, whose components are i.i.d., uniformly distributed on

[−∆/2,∆/2), and independent of xj[k]. In the presence of small quantization

noise, we can appeal to a first-order Taylor approximation of the vector function

g(·), and approximate the updating rule (3.104) as:

xi[k + 1] ≃ xi[k] + α[k]

[

−∇xi[k]Ii(xi[k]) +M∑

j=1

aij [k] g(xj [k]− xi[k]) +

− ηi[k] +

M∑

j=1

aij[k] Jg(xj [k]− xi[k])(ν ij[k] + ǫij [k])

]

,

i = 1, . . . ,M, (3.105)

142


where Jg(xj [k]−xi[k]) is the Jacobian matrix of g(·) evaluated at (xj[k]−xi[k]).

Now, exploiting the structure of the function g(·) in (3.4) and the features of linear

attraction in (3.7) and bounded repulsion in (3.8), the recursion (3.105) can be

expressed as:

xi[k + 1] ≃ xi[k] + α[k]

[

−∇xi[k]Ii(xi[k]) +

+

M∑

j=1

aij [k]

[

cA − cR exp

(

−‖xj [k]− xi[k]‖

2

cG

)]

(xj [k]− xi[k]) +

− ηi[k] +

M∑

j=1

aij [k] Jg(xj [k]− xi[k])(ν ij[k] + ǫij [k]),

i = 1, . . . ,M. (3.106)

To rewrite (3.106) in compact form, we introduce the random vectors Υx[k] and

Ψx[k] ∈ RnM with vector components

[

Υx[k]]

i=

M∑

j=1

aij [k]Jg(xj [k]− xi[k])ν ij[k], (3.107)

[

Ψx[k]]

i=

M∑

j=1

aij [k]Jg(xj [k]− xi[k])ǫij[k]. (3.108)

The vectors Υx[k] and Ψx[k] are the state dependent aggregated contribution of

quantization and dithering. It follows from the conditions on the dither, that

E[Υx[k]] = E[Ψx[k]] = 0, ∀k,

supx,k

[

‖Υx[k]‖2]

= supx,k

[

‖Ψx[k]‖2]

≤M(M − 1)n∆2λmax(Jg)

12,

from which we get

supx,k

[

‖Υx[k] +Ψx[k]‖2]

≤ 2 supx,k

[

‖Υx[k]‖2]

+ 2 supx,k

[

‖Ψx[k]‖2]

≤M(M − 1)n∆2λmax(Jg)

3= ζq. (3.109)

143


with λmax(Jg) = maxx λmax(JTg (x)Jg(x)) > 0. To prove the validity of the

bound in (3.109), we have still to show that λmax(Jg) is finite or upper bounded

by a finite value. The choice of the coupling function g(·) in (3.7)-(3.9), with

linear attraction and bounded repulsion, ensures the boundedness of the partial

derivatives in Jg(x). In the scalar case, this is straightforward to see. Indeed, in

this case, we have

g(y) = [cA − cR exp(−y2/cG)]y (3.110)

⇒ g′(y) = cA − cR exp(−y2/cG)(1− 2y2/cG), (3.111)

and it is easy to see that g′(y) can assume only bounded values. In the vector case,

at the same way, we can guarantee that the elements of Jg(x) take values from

a finite set. Then, as a consequence of the Gershgorin circles theorem [178], the

eigenvalues of the matrix JTg (x)Jg(x) assume bounded values. Let Jmax denote

the maximum value (in modulus) assumed by the elements of JTg (x)Jg(x), ∀

x. Then, by the Gershgorin theorem, an upper bound for λmax(Jg) is given by

n2M2J2max.

The overall evolution dynamics can then be expressed in compact form as:

x[k + 1] = x[k] + α[k][

−Σ′(x[k])−(

Lx[k]⊗ In)

x[k] +Υx[k] +

+ Ψx[k]−Ξ[k]]

, (3.112)

where Υx[k] and Ψx[k] are the state dependent aggregated contribution of quan-

tization noise in (3.107) and (3.108), Σ′(x[k]) = col[∇xi[k]Ii(xi[k])]i=1,...,M , and

Ξ[k] = col[ηi[k]]i=1,...,M is the overall estimation noise vector. The state-dependent

Laplacian matrix is given by

Lx[k] =Dx[k]−Ax[k] (3.113)

whereAx[k] denotes a symmetric state-dependent adjacency matrix whose entries

are given by

[

Ax[k]]

ij= aij [k]

[

cA − cR exp(

−‖xj [k]−xi[k]‖2

cG

)]

,[

Ax[k]]

ii= 0.

(3.114)

144


We consider four assumptions on the stochastic procedure (3.112):

Assumption B.1 : (Estimation noise) We assume that the observation noise

process Ξ[k] = colηi[k]i=1,...,M in (3.102) is an i.i.d. zero mean process, with

finite second order moment, i.e.,

E[Ξ[k]TΞ[k]] ≤ ϕe, for all k. (3.115)

Assumption B.2 : (Independence) The sequences Lx[k]k≥0, Υx[k]k≥0,

Ψx[k]k≥0 and Ξ[k]k≥0 are mutually independent.

Assumption B.3 : (Markov) Consider the filtration Fxk k≥0, given by

Fxk = σA

(

x(0), Lx[n],Υx[n],Ψx[n],Ξ[n]0≤n<k

)

(3.116)

where σA(·) denotes sigma algebra. It then follows that the random quantities

Lx[k], Υx[k], Ψx[k] and Ξ[k] are independent of Fxk , implying that x[k],Fx

k k≥0

is a Markov process.

Assumption B.4 : (Persistence) To obtain convergence, we assume that the

step size α[k] satisfies the following conditions:

α[k] > 0,

∞∑

k=0

α[k] =∞,∞∑

k=0

α2[k] <∞. (3.117)

Condition (3.117) ensures that the step-size decays to zero, but not too fast. An

example of step-size sequence that satisfies (3.117) is

α[k] =α0

(k + 1)β, α0 > 0, 0.5 < β ≤ 1. (3.118)

The following theorem presents a classical result from stochastic approxima-

tion theory from [183] regarding the convergence properties of generic stochastic

recursive procedures; the result will be used to establish the convergence of the

swarming algorithm.

145


Theorem 13 Let x[k]k≥0 be a random vector defined by the difference equation

x[k + 1] = x[k] + α[k][

R(x[k]) + Γ(k,x[k], ω)]

(3.119)

with initial condition x[0] = x0, where R(·) : Rn → R

n is Borel-measurable,

Γ(k,x[k], ω) is a family of zero-mean random vectors in Rn, defined on some

probability space (Ω,F ,P), and ω ∈ Ω is a canonical element of Ω. Consider the

following set of conditions:

Condition C.1 : The function Γ(k, ·, ·) : Rn ×Ω→ Rn is Bn ⊗F measurable 3

for all k.

Condition C.2 : There exists a filtration Fkk≥0 of F , such that, for every

k, the family of random vectors Γ(k,x[k], ω)x∈Rn is Fk measurable and indepen-

dent of Fk−1. (Under the conditions C.1 and C.2, the random vector sequence

x[k]k≥0 is a Markov process.)

Condition C.3 : There exists a nonnegative function V (x) ∈ C2 with bounded

second-order partial derivatives satisfying the conditions

lim‖x‖→∞

V (x) = ∞ (3.120)

supx∈Uǫ,1/ǫ(S)

< R(x),∇xV (x) > < 0 for ǫ > 0 (3.121)

where < ·, · > denotes the inner product operator, S = x : R(x) = 0 is the

solution set and Uǫ,1/ǫ(S) = x ∈ Rn : ǫ < ‖x− xs‖ < 1/ǫ,xs ∈ S, ǫ > 0.

Condition C.4 : There exist a constant K > 0, such that,

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ K(1 + V (x)). (3.122)

Condition C.5 : The step size sequence α[k]k≥0 satisfies (3.117).

Let the conditions C.1-C.5 hold for the process x[k]k≥0. Then, x[k]k≥0 is a

Markov process and, starting from an arbitrary initial condition x0, it converges

3B

n denotes the Borel algebra of Rn.

146


almost surely (a.s.) as k → ∞, either to a point of the solution set S = x :

R(x) = 0, or to the boundary of one of its connected components.


In the following, we will use Theorem 1 to establish the a.s. convergence of

the iterative swarming procedure (3.112). By decomposing the state dependent

Laplacian matrix Lx[k] into the sum of a mean part plus a random part as in

(3.97), expression in (3.112) can be written as:

x[k + 1] = x[k] + α[k][

−Σ′(x[k])−(

Lx ⊗ In)

x[k]−(

Lx[k]⊗ In)

x[k] +

+ Υx[k] +Ψx[k]−Ξ[k]]

. (3.123)

In the notation of Theorem 1, equation (3.123) can be recast as in (3.119), where

R(x[k]) = −Σ′(x[k])−(

Lx[k]⊗ In)

x[k], (3.124)

Γ(k,x[k], ω) = −(

Lx[k]⊗ In)

x[k] +Υx[k] +Ψx[k]−Ξ[k]. (3.125)

In this way, the original swarming problem has been converted into the search

for the zeros of the deterministic function R(x[k]), whose value is measurable

at each time instant k and corrupted by an additive zero-mean random distur-

bance Γ(k,x[k], ω). We are now able to state the main theorem of the swarming

behavior in the presence of random disturbances.

Theorem 14 Consider the discrete swarming algorithm in (3.104) with arbitrary

initial state x0. Under the hypothesis of a small additive quantization noise and

the assumptions A.1-A.3 and B.1-B.4, the algorithm converges almost surely

(a.s.) as k →∞ to one of the zeros of the function R(x) in (3.124) or, equiva-

lently, to a local minimum of the function J(x) in (3.2) evaluated for the expected

graph. Then,

P

[

limk→∞

ρ(x[k], S) = 0

]

= 1 (3.126)

where ρ(·) is the standard Euclidean metric norm and S = x : R(x) = 0 is the

solution set.

147


Proof. The proof follows by showing that the process x[k]k≥0, generated by

the swarming algorithm, satisfies the conditions C.1-C.5 of Theorem 1. Recall

the filtration given in equation (3.116). Under the Assumptions B.1-B.4, the

random family Γ(k + 1,x[k], ω) is Fxk+1 measurable, zero mean and independent

of Fxk . As a consequence, the conditions C.1, C.2 of Theorem 2 are satisfied

and the random vector sequence x[k]k≥0 is a Markov process. We will show

now the existence of a stochastic potential function V (x) such that the swarming

algorithm in (3.112) satisfies the conditions C.3, C.4. To this end, we define

V (x) = tr(Σ(x)) + cAxT (L⊗ In)x+ cGcR x

T (Lr,x ⊗ In)x (3.127)

which coincides with the system potential function in (3.89) evaluated for the

expected graph Laplacian L. As shown for the expression in (3.89), V (x) ∈ C2 is

a nonnegative function and, under the profile’s smoothness assumption A.1 and

the choice of attraction and repulsion functions in (3.7) and (??), it has bounded

second order partial derivatives. Under Assumption A.3, the vector x lies on a

subspace orthogonal to N(

L⊗In)

, where N (·) denotes the nullspace of a matrix.

Hence, applying (2.80) and considering the expression of λ(L⊗ In) in (2.83), we

find that the expression (3.127) admits the lower bound:

V (x) ≥ tr(Σ(x)) + cAλ2(L)‖x‖2 + cGcR x

T (Lr,x ⊗ In)x. (3.128)

It is then straightforward to see how the potential function V (x) satisfies condi-

tion (3.120). Since the gradient of V (x) is given by

∇xV (x) = Σ′(x) +(

Lx ⊗ In)

x, (3.129)

it then follows that

< R(x),∇xV (x) > = −[

Σ′(x) +(

Lx ⊗ In)

x]T [

Σ′(x) +(

Lx ⊗ In)

x]

= −‖R(x)‖2 < 0 for all x 6= xs ∈ S. (3.130)

Thus, condition C.3 of Theorem 2 is satisfied. From equation (3.124), applying

148


the Cauchy-Schwartz inequality, we get the following upper bound

‖R(x)‖2 =∥

∥−Σ′(x)−(

Lx ⊗ In)

x∥

∥

2

= ‖Σ′(x)‖2 + 2Σ′(x)T(

Lx ⊗ In)

x+ ‖(

Lx ⊗ In)

x‖2

≤ ‖Σ′(x)‖2 + 2‖Σ′(x)‖‖(

Lx ⊗ In)

x‖+ ‖(

Lx ⊗ In)

x‖2. (3.131)

Under assumption A.1, we have ‖Σ′(x)‖ ≤ σM , and the previous bound can be

recast as

‖R(x)‖2 ≤ σ2M + 2σM |λM (Lx)|‖x‖+ λ2M (Lx)‖x‖2 (3.132)

where λM (Lx) is the maximum (in modulus) eigenvalue of Lx. The state depen-

dent mean Laplacian Lx = Dx−Ax in (3.113) depends on x through a bounded

function. Hence, for any values of the swarm constants cA, cR and cG, it always

exists a constant c1 > 0 such that |λM (Lx)| ≤ c1 and

‖R(x)‖2 ≤ σ2M + 2σMc1‖x‖+ c21‖x‖2

= c2 + c3‖x‖+ c4‖x‖2 (3.133)

where c2 = σ2M > 0, c3 = σMc1 > 0 and c4 = c21 > 0. Now, adding the function

c3/2(1 − ‖x‖)2 ≥ 0, the previous bound gives

‖R(x)‖2 ≤ c2 + c3/2 + (c4 + c3/2)‖x‖2 = c5 + c6‖x‖

2 (3.134)

where c5 = c2 + c3/2 > 0 and c6 = c4 + c3/2 > 0.

From equation (3.125) and the independence assumption B.2,

E‖Γ(k,x, ω)‖2 = E∥

∥−(

Lx[k]⊗ In)

x+Υx[k] +Ψx[k]−Ξ[k]∥

∥

2

= xTE[(

Lx[k]⊗ In)T

(Lx[k]⊗ In)]

x+

+ E‖Υx[k] +Ψx[k]‖2 + E‖Ξ[k]‖2. (3.135)

The eigenvalues of the Laplacian error Lx are bounded because this matrix takes

values from a finite set and it depends on the state x through a bounded function.

149


Then, considering the bounds in (3.109) and (3.115), we get

E‖Γ(k,x, ω)‖2 ≤ E[

λM (Lx[k]T Lx[k])

]

‖x‖2 + ζq + ϕe

= c7 + c8‖x‖2 (3.136)

where c7 = ζq + ϕe > 0 and c8 = maxx E[

λM (LTxLx)

]

> 0. We then have from

(3.134) and (3.136)

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ c9 + c10‖x‖2 (3.137)

where c9 = max(c5, c7) > 0 and c10 = max(c6, c8) > 0. Exploiting now Assump-

tion A.3, the vector x lies on a subspace orthogonal to N(

L⊗In)

and, under the

assumption that the repulsion force is strong enough to avoid the overall collapse

of the swarm onto the center, the overall consensus over x is never reached. This

means that the inequality 0 < λ2(L)‖x‖2 ≤ xT

(

L ⊗ In)

x holds for all x, and,

substituting it in (3.137), we get

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ c9 +c10λ2(L)

xT(

L⊗ In)

x

= c9 + c11xT(

L⊗ In)

x, (3.138)

where c11 = c10/λ2(L) > 0. Summing now the potential function V (x) ≥ 0 in

(3.127) to the last expression, we can write the bound

‖R(x)‖2 + E‖Γ(k,x, ω)‖2 ≤ c9 + c11xT(

L⊗ In)

x+ V (x)

≤ c9 + tr(Σ(x)) + (cA + c11)xT(

L⊗ In)

x+ cGcRxT(

Lr,x ⊗ In)

x

= c9 + tr(Σ(x)) + c12cAxT(

L⊗ In)

x+ cGcRxT (Lr,x ⊗ In

)

x

≤ c9 + c12V (x) ≤ K(1 + V (x)) (3.139)

where c12 = (1 + c11/cA) > 1 and K = max(c9, c12) > 0. This verifies also condi-

tion C.4 of Theorem 1 and condition C.5 is satisfied by the choice of α[k]k≥0

made in the B.4. This concludes our proof.

150


Fast Swarming Algorithms

As in the preceding section, we can consider a general scaled gradient opti-

mization in order to improve the performance of the algorithm. Considering the

presence of random disturbances, the discrete time version of a general scaled

gradient optimization can be recast in compact form as:

x[k + 1] = x[k] + α[k]B(x[k])[

−Σ′(x[k])−(

Lx[k]⊗ In)

x[k] +Υx[k]

+ Ψx[k]−Ξ[k]]

. (3.140)

where B(x[k]) = diag(fi(Ii(xi(k)))In)i=1,...,M . In the notation of Theorem 1,

the equation (3.140) can be written as in (3.119) where

R(x[k]) = −B(x[k])

[

Σ′(x[k]) +(

Lx[k]⊗ In)

x[k]

]

(3.141)

Γ(k,x[k], ω) = −B(x[k])

[

(

Lx[k]⊗ In)

x[k]−Υx[k]−Ψx[k] +Ξ[k]

]

(3.142)

The following convergence result holds for the adaptive swarming in (3.140).

Theorem 15 Consider the discrete swarming algorithm in (3.140) with arbitrary

initial state x0. Under the hypothesis of a small additive quantization noise and

the assumptions A.1-A.2 and B.1-B.4, the algorithm converges a.s. as k →∞

to one of the zeros of the function R(x) in (3.142) or, equivalently, to a local

minimum of the function J(x) in (3.2) evaluated for the mean graph. That is

P

[

limk→∞

ρ(x[k], S) = 0

]

= 1 (3.143)

where ρ(·) is the standard Euclidean metric norm and S = x : R(x) = 0 is the

solution set.

Proof. The proof follows the same steps as in Theorem 2. Under assumption

B.3, the sequence generated by the swarming algorithm in (3.140) is a Markov

process. We consider again the nonnegative function V (x) in (3.127). SinceR(x)

in (3.142) is a scaled gradient descent direction for the optimization of V (x), it

151


is easy to show that the Lyapunov condition in (3.121) is always verified for all

x outside the solution set S. Applying now the assumptions A.1, A.3, B.1 and

B.2 and considering the boundedness of the eigenvalues of the diagonal matrix

B(x), some algebra shows that the inequality in (3.122) holds, thus concluding

the proof.

Remark: The matrix B(x) in (3.140) is a full rank matrix. Hence, the zeros of

the function R(x) in (3.142) coincide with those of the function in (3.124).

In the following sections, we will present numerical results illustrating how the

proposed algorithm (3.140) outperforms the basic swarming algorithm (3.123) in

terms of convergence speed and resilience against channel imperfections.

Projected Methods

In the previous section, we showed how the gradient of the sum-rate is affected

by the randomness introduced by the link failures and by the quantization error

present on the data exchanged between SUs. In this section, we consider the dis-

tributed solution of the constrained optimization problem (3.94), which leads to

a projected swarm method. To find a solution of the problem (3.94) affected by

random disturbances, stochastic approximation algorithms assume relevance. In

the remainder of this section we introduce a stochastic approximation scheme for

solving the problem in (3.94) in a distributed manner. In particular, we consider

a projection-based Robbins-Monro (RM) stochastic approximation procedure.

The problem is amenable for distributed solutions because the optimization set

X =∏M

i=1Xi is given by Cartesian product of sets Xi, allowing the parallel com-

putation of the algorithm. In particular, we consider a projection-based swarming

algorithm where at each time k, every SU simultaneously updates its resource

allocation vector according to

x[k + 1] = [x[k] + α[k]∇xJ(x[k])]X = T (x[k]), (3.144)

k ≥ 0, q = 1, . . . , Q,

152


where α[k] is an iteration-dependent step size, and ∇xJ(x[k]) ∈ RnM is the

discrete-time version of (3.3). As shown in the previous section, due to the effect

of the random link failures and quantization, the gradient of the potential function

in (3.89) can be written as:

∇xJ(x[k]) = Σ′(x[k])−(

Lx[k]⊗ In)

x[k] +Υx[k] +Ψx[k]−Ξ[k]

= R(x[k]) + Γ(k,x[k], ω), (3.145)

with R(x[k]) and Γ(k,x[k], ω) given by (3.124) and (3.125), respectively. To

prove the convergence of the iterative procedures in (3.144), we use a known

result from supermartingale theory, which we provide for convenience.

Lemma 5 Let Y [k], W [k], Z[k] be three sequences such that Wk is non-negative

for all k. If almost surely∑∞

k=1 Z[k] <∞, and

Y [k + 1] ≤ Y [k]−W [k] + Z[k], k ≥ 0, (3.146)

then, almost surely, either Y [k] → −∞ or else Y [k] converges to a finite

value and∑∞

k=1W [k] <∞.

Proof. The proof can be found in [174], Lemma 1.

It follows the convergence result on the projection-based swarming algorithm in

the presence of random disturbances.

Theorem 16 Let x[k] be the sequence generated by the distributed stochastic

swarming algorithm in (3.144), with step-size satisfying the conditions in (3.117).

Then, the potential sequence J(x[k])k≥0 converges almost surely to a finite value

J∗, i.e.,

Prob

[

limk→∞

J(x[k]) = J∗

]

= 1, (3.147)

where Prob[E ] denotes the probability of the event E. Furthermore, let x∗ be an

accumulation point of the sequence x[k], almost surely the optimal local solution

x∗ is a fixed point of the mapping T (x) = colT i(x)Mi=1, such that x∗ = T (x∗).

153


Proof. To prove the convergence of the sum-rate J(x), we use the following

Lemma.

Lemma 6 (Descent Lemma [189]): If F : RN → R is continuously differentiable

and its gradient is Lipschitz continuous, i.e.,

‖∇F (x)−∇F (y)‖2 ≤ L‖x− y‖2 ∀x,y ∈ RN (3.148)

then,

F (x+ y) ≤ F (x) + yT∇F (x) +L

2‖y‖22 ∀x,y ∈ R

N . (3.149)


In the k-th iteration of the algorithm, each SU simultaneously updates its resource

allocation. One sufficient condition for Lipschitz continuity is that the L2-norm

of the Hessian matrix of J(x) is bounded, in which case this bound can be used

for the Lipschitz constant L. Considering the choice of a coupling function g(·)

with linear attraction (3.7) and bounded repulsion (3.10) and resorting to the

assumption A.1, it can be shown this is true for J(x) and, specifically, there

exists a positive constant L which upper bounds the L2-norm of the Hessian

matrix of J(x). Applying the descent Lemma to J(x), we get

J(x[k+1]) ≤ J(x[k])+[x[k+1]−x[k]]T∇xJ(x[k])+L

2‖x[k+1]−x[k]‖22 (3.150)

Now, we recall some basic properties of projection mappings from [189]:

1. For every x ∈ RN , there exists a unique z ∈ X that minimizes ‖z − x‖2

over all z ∈ X, and will be denoted as [x]X ;

2. Given some x ∈ RN , a vector z ∈ X is equal to [x]X if and only if (y −

z)T (x− z) ≤ 0 for all y ∈ X;

3. The mapping f : RN → X defined by f(x) = [x]X is continuous and

nonexpansive, that is, ‖[x]X − [y]X‖ ≤ ‖x− y‖2 for all x,y ∈ RN .

154


Applying property 2 of the projection mapping and using the expression in

(3.144), we have

[x[k]− x[k + 1]]T (x[k]− α[k]∇xJ(x[k])− x[k + 1]) ≤ 0 (3.151)

which yields α[k][x[k+1]−x[k]]T∇xJ(x[k]) ≤ −‖x[k+1]−x[k]‖22. Substituting

the last expression in (3.150), we get

J(x[k + 1]) ≤ J(x[k])−1

α[k]‖x[k + 1]− x[k]‖22 +

L

2‖x[k + 1]− x[k]‖22. (3.152)

In the notation of Lemma 1, expression (3.152) can be recast as in (3.146), where

Y [k] = J(p[k]), W [k] = (1/α[k])‖x[k + 1]− x[k]‖22,

Z[k] = (L/2)‖x[k + 1]− x[k]‖22. (3.153)

By choosing a positive step size α[k], the sequence W [k] is nonnegative for all

k. Moreover, exploiting the expression in (3.144) and the fact that x[k] ∈ X , we

have

Z[k] =L

2‖x[k + 1]− x[k]‖22 =

L

2‖[x[k] + α[k]∇xJ(x[k])]X − [x[k]]X ‖

22 .

Now, using the non-expansivity of the projection operator and substituting the

expression in (3.145), we get

Z[k] ≤L

2‖α[k]∇xJ(x[k])‖

22 =

L

2α2[k] ‖R(x[k]) + Γ(k,x[k], ω)‖22

≤ Lα2[k][

‖R(x[k])‖22 + ‖Γ(k,x[k], ω)‖22

]

. (3.154)

SinceR(x[k]) is the gradient of J(x) with respect to x, evaluated for the expected

graph L, the resource allocation vector lies into a bounded set, and considering

Assumption A.1, the function ‖R(x[k])‖22 can be upper bounded by a positive

constant C2, for all k. To proceed with the proof, we will use the following lemma.

Lemma 7 Let b[k] be a sequence of random variables with each b[k] being Fk+1

measurable, and suppose that E[b[k]|Fk] = 0 and E[‖b[k]‖2|Fk] ≤ B, where B is

155


some deterministic constant. Then, the sequences

∞∑

k=0

α[k]b[k] and

∞∑

k=0

α2[k]‖b[k]‖2 (3.155)

converge to finite limits (with probability 1), if the step size α[k] satisfies condi-

tions (3.117).

Proof. Since∑∞

k=0 α[k]b[k] is a martingale whose variance is bounded by the

series B∑∞

k=0 α[k], it must have a finite limit by the martingale convergence

theorem. Furthermore,

E

[

∞∑

k=0

α[k]2‖b[k]‖2

]

≤ B∞∑

k=0

α2[k], (3.156)

showing that∑∞

k=0 α2[k]‖b[k]‖2 converges to a finite limit with probability 1.

Now, considering the expression of Γ(k,x[k], ω) in (3.125), we have from

(3.136) that

E[‖Γ(k,x[k], ω)‖Fk ]2 ≤ C3 +C4‖x‖

2, (3.157)

and, since the resource allocation vector lies into a bounded set, we have ‖x‖2 ≤

C5. As a consequence of the previous bound, we get

E[‖Γ(k,x[k], ω)‖2Fk] ≤ C3 + C4C5 = C6 > 0. (3.158)

Now, resorting to the bound on the sequence Z[k] and on ‖R(x[k])‖22, we have

∞∑

k=0

Z[k] ≤∞∑

k=0

Lα2[k][

‖R(x[k])‖22 + ‖Γ(k,x[k], ω)‖22

]

≤ LC2

∞∑

k=0

α2[k] + L

∞∑

k=0

α2[k]‖Γ(k,x[k], ω)‖22. (3.159)

Exploiting the conditions in (3.117), the first term of the right hand side of (3.159)

has a finite limit. Moreover, due to the fact that E[Γ(k,x[k], ω)|Fk ] = 0 and

E[‖Γ(k,x[k], ω)‖2|Fk] ≤ C6, for all k, lemma 3 applies and also the second term

156


of the right hand side of (3.159) has a finite limit. Then, we conclude that the

series∑∞

k=0 Z[k] <∞, having a finite limit with probability 1. All the conditions

of Lemma 1 are then satisfied and the result applies. Hence, almost surely, either

Y [k] → −∞ or else Y [k] converges to a finite value and∑∞

k=0W [k] < ∞.

Since the function J(x) is bounded from below, the sequence Y [k] = J(x[k])

cannot diverge to −∞ and it has to converge almost surely to a finite value.

Furthermore, almost surely it must be that

∞∑

k=1

W [k] =

∞∑

k=0

1

α[k]‖x[k + 1]− x[k]‖22 =

∞∑

k=0

1

α[k]‖T (x[k])− x[k]‖22 <∞,

(3.160)

where T (x[k]) = col [T i(x[k])]Mi=1. However, if the step size α[k] satisfies (3.117),

the sequence 1/α[k] is divergent. Then, since∑∞

k=1W [k] <∞, it must follow

limk→∞

‖T (x[k])− x[k]‖22 = 0 a.s. (3.161)

Then, if x∗ is an accumulation point of the sequence, we can a.s. guarantee

subsequence convergence such that T (x∗) = x∗. This concludes our proof.

Numerical Examples

In this section, we provide numerical examples to illustrate the performance

and main features of the proposed swarming techniques impaired by channel

randomness.

Numerical Example 1 - Swarming in the Presence of Link Failures, Quantization

and Estimation Errors: The purpose of this first example is to show the per-

formance of the proposed allocation algorithm in the presence of random packet

drops, errors in the estimation of the profile gradient and quantization noise. We

consider a connected network composed of 15 SUs, plus two PUs. The topology of

the network corresponding to the case in which all packets are correctly delivered

is shown in Fig. 3.19, where the SUs are represented by dots, while the PUs are

indicated by squares. We consider two examples of interference profiles (supposed

157


0 0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PU

PU

Figure 3.19: Secondary network. The square nodes denote primary users and the

circle nodes denote secondary users.

to be the same for all the nodes), as shown in Fig. 3.20, where the the black curve

represents the true spectrum, whereas the red lines report the noisy observation.

The number of resources (frequency subchannels) to be allocated is assumed to be

15, equal to the number of cognitive users. The resources are initially scattered

randomly across the frequency spectrum. At the k-th iteration of the updating

rule (3.104), each node communicates to its neighbors the position it intends to

occupy, i.e., the scalar xi[k] representing a frequency subchannel. Because of fad-

ing and additive noise, a communication link among two neighbors has a certain

probability p to be established correctly. The values to be exchanged are also af-

fected by quantization noise, supposed to be small with respect to the bandwidth

of the frequency subchannel. The error in the estimation of the profile gradient

is assumed to be Gaussian distributed with zero mean and variance σ2e = 1. Two

examples of resource allocation are shown in Fig. 3.20, where the dots represent

the final frequency channels chosen at convergence by the network nodes. The

parameters of the swarm are cA = 0.02, cR = 0.5, cG = 1 and we considered

158


20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25−0.5

0

0.5

1

1.5

Frequency (Mhz)

PS

D (

mW

/Hz)

30 35 40 45−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Frequency (Mhz)

PS

D (

mW

/Hz)

Figure 3.20: Examples of resource allocation by swarming.

p = 0.7. In both cases, it is evident how the resources avoid the position occu-

pied by primary users, tend to keep the spread as small as possible while avoiding

collisions among the allocations of different users. Observe that the number of

allocated channels is less than the number of requested resources. This means

that a certain number of nodes have picked up the same channels. We have

checked numerically that, by choosing appropriately the swarm parameters, the

final channel allocation does not lead to collisions among spatial neighbors. This

means that the algorithm is capable of implementing a decentralized mechanism

for spatial reuse of frequencies.

To show the effect of link failures on resource allocation, in Fig. 3.21 we

159


0 50 100 150 2000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Iteration index

Avera

ge Inte

rfere

nce

Ideal Case

p = 0.7

p = 0.5

p = 0.3

Figure 3.21: Average interference perceived by the swarm vs. time index, for

different probabilities of correct packet reception.

report the average interference level perceived by the swarm versus the iteration

index, averaged over 100 independent realizations of random drops. We consider

different probabilities p of correct packet reception; the ideal case, which corre-

sponds to p = 1, is shown as a benchmark. The interference profile is the one

shown in the left side of Fig. 3.20; the network topology is the one depicted in

Fig. 5.1, the swarm parameters are cA = 0.02, cR = 0.5, cG = 1. The iteration

dependent step size is given by α[k] = α0/k, with α0 = 0.5, in order to satisfy

(3.117). From Fig. 5.1 we notice that, after sufficient time, the network always

reaches an equilibrium state that coincides with a swarm cohesion in the low

interference region of the spectrum. Interestingly, we can observe that the final

interference level is always the same, independently of the link failure probability

1−p. From Fig. 3.21, we see that the only effect of the random link failures is to

slow down the convergence, but without affecting the final average interference

level perceived by the swarm. This observation illustrates the robustness of the

proposed algorithm.

160


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

Probability to establish a link

Avera

ge Inte

rfere

nce

cA = 0.01

cA = 0.02

cA = 0.03

Figure 3.22: Average interference perceived by the swarm at convergence, versus

the probability to establish a communication link, for different values of the swarm

attraction parameter cA.

In some applications, the number of iterations must be limited to avoid ex-

cessive delays. It is then of interest to assess the performance of the distributed

resource allocation strategy, fixing a maximum number of iterations Nmax. To

this end, in Fig. 3.22 we report the average interference level perceived by the

swarm, versus the probability p to establish a communication link, averaged over

the frequency slots occupied by the SUs, after Nmax = 50 iterations. The results

are averaged over 100 independent realizations, considering three different values

of the swarm attraction parameter cA. The other parameters of the swarm are

cR = 0.5, cG = 1; the iteration dependent step size is chosen as before, with

α0 = 0.1. From Fig. 3.22, we notice that, for that given number of iterations, at

low values of the probability p, the interference perceived by the swarm is high

because some nodes are trapped in regions occupied by the primary users and

characterized by a low gradient profile. This happens because the duration of the

161


20 30 40 50 60 70 80 90 10020

40

60

80

100

120

140

160

180

Number of nodes

Avera

ge c

onverg

ence t

ime (

itera

tions)

nb = 3

nb = 4

nb = 5

Ideal case

Figure 3.23: Average convergence time versus number of nodes, for different

number of bits used for quantization.

iterative algorithms has not been sufficient to move out the resources trapped

in wrong locations. However, as p increases, the perceived interference decreases

because the attraction exerted by the swarm is finally able to move the resources

toward the interference-free region.

What is also interesting to observe from Fig. 3.22 is that, increasing the at-

traction parameter, for any given p, the performance of the resource allocation

improves. This example shows that the cohesion force represents an intrinsic

robustness factor of the algorithm. In fact, resources allocating over high in-

terference bands might measure a flat spectrum, thus resulting in limited capa-

bilities to move out of (flat) occupied bands, if the only cause of change is the

spectrum gradient. However, increasing the cohesion force, the agents allocating

over the low interference band tend to form cohesive blocks that exert an attrac-

tion towards the agents trapped by mistake over the flat regions of the spectrum

occupied by the primary users. This is an example of cooperation gain.

Finally, it is of interest to check the effect of network size and quantization

162


20 30 40 50 60 70 80 90 10020

40

60

80

100

120

140

160

Number of nodes

Avera

ge c

onverg

ence t

ime (

itera

tions)

connectivity parameter β = 1.1



Figure 3.24: Average convergence time versus number of nodes, for different

degrees of network connectivity.

noise on the convergence rate. In Fig. 3.23, we report the average number

of iterations needed by the algorithm to converge versus the number of nodes

composing the network, averaged over 200 different network configurations, con-

sidering different numbers of bits to quantize the exchanged messages. The ideal

case, which corresponds to absence of quantization, is shown as a benchmark.

The network graph is a random geometric graph with node’s covering radius

r0 = β√

2 log(M)/(πM), where β = 1.3 tunes the average network connectivity.

The swarm parameters are cA = 0.2, cR = 1, cG = 1 and the initial step size is

α0 = 0.5. As expected, from Fig. 3.23, we can notice how, increasing the number

of nodes and reducing the number of bits used for quantization, the algorithm

needs more time to converge. In a noise-free case, a higher network connectivity

implies faster convergence, whereas, in the noisy case, the algorithm needs more

time to converge due to the larger variance of the disturbance that affects the

system. To show this behavior, in Fig. 3.24, we repeat the previous simulation,

fixing the number of quantization bits to nb = 4 and considering different val-

163


0 20 40 60 80 100 1200.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Iteration index

Avera

ge Inte

rfere

nce

Basic Swarming − Ideal Case

Basic Swarming − p = 0.5

Adaptive Scaling − Ideal Case

Adaptive Scaling − p = 0.5

Figure 3.25: Average interference perceived by the swarm vs. time index, for

different algorithms and probabilities of correct packet reception.

ues of the network connectivity parameter β. As we can notice from Fig. 3.24,

increasing the network connectivity, the algorithm needs more time to converge.

Numerical Example 2 - Effect of Adaptation on Performance: In this example we

aim to show the benefits achievable by introducing adaptation and learning as

in (3.140). We consider linear scaling functions fi(Ii(xi(k))) = ai + biIi(xi(k)),

where ai = 0.1 (for all i) and the slope parameter bi is chosen in order to increase

the convergence speed of the nodes perceiving a high interference. We assume

the presence of the interference profile shown in the left side of Fig. 3.20 and

the network topology depicted in Fig. 5.1. In the first example we compare the

convergence speed of the gradient based swarming algorithm and the adaptive

method in (3.140) with scaling coefficients weighted by the perceived interference

power. In Fig. 3.25, we report the average interference level perceived by the

swarm, averaged over 100 independent realizations, versus the iteration index,

considering two different values of probability p to establish a link. In the simu-

164


2 4 6 8 10 12 14 16 18 200.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

Slope parameter b

Avera

ge Inte

rfere

nce

cA = 0.02 p = 0.9

cA = 0.02 p = 0.5

cA = 0.03 p = 0.9

cA = 0.03 p = 0.5

Figure 3.26: Average interference perceived by the swarm at convergence, versus

the slope parameter of the linear scaling functions, for different probabilities of

correct packet reception and different values of the attraction parameter cA.

lation, we consider a linear scaling function with parameters ai = 0.1 and bi = 2

for all i. The parameter of the swarm are cA = 0.02, cR = 0.5, cG = 1 and the

step size is chosen such that α0 = 0.5 for both the algorithms. The simulation

shows how the swarming algorithm with adaptive scaling is robust with respect

to link failures, achieving the same performance level of the basic algorithm and

greatly outperforming it in terms of convergence speed. This means that the

convergence time of the swarming algorithm can be considerably reduced if every

node adapts its convergence speed according to the perceived interference.

To measure the effectiveness of the distributed resource allocation strategy

in the presence of a limited maximum number of iterations, in Fig. 3.26 we

report the interference level, versus the slope parameter bi of the linear scaling

functions, averaged over the frequency slots occupied by the SUs, after Nmax =

20 iterations. The result is averaged over 100 independent realizations. We

165


considered two different values of the probability p and of the swarm attraction

parameter cA. The other swarm parameters are equal to cR = 0.5 and cG = 1.

From Fig. 3.26, we notice that, at low values of the parameter bi, the movement

of the resources is very limited and, inside the maximum number of iterations,

some resources cannot move out form the regions occupied by the primary users

because of the random impairments affecting the algorithm. As bi increases, the

resources perceiving a high power move faster toward the interference-free region

due to the increment of the average profile gradient and the cohesion force, thus

making the overall swarm experience a smaller total interference. This means

that the performance of the swarming algorithm can be considerably improved

if every node adapts its scaling function according to the perceived interference.

Furthermore, from Fig. 3.26, we notice how an increment of the cohesion force

induces a better performance thanks to the cooperation among network nodes,

similarly to what we had observed from Fig. 3.22. From Fig. 3.26, we also

notice, as expected, how a lower probability to establish a communication link

determines worst performance.

Numerical Example 3 - Distributed Graph Coloring: One of the most interesting

features of the proposed swarming technique is its capability to induce a spatial

reuse of frequency channels from secondary nodes far away from each other, using

a decentralized approach under random packet dropping. As an example of chan-

nel allocation, in Fig. 3.27 we consider a network composed of 50 nodes, where

each node senses the interference profile shown in the left side of Fig. 3.20. The

swarm parameters are cA = 0.01, cR = 0.5 and cG = 1. Every node starts from a

random initial position on the spectrum, and then it updates its intended position

according to (3.104), where the probability to establish correctly a communica-

tion link between secondary users is p = 0.7. In the application at hand, there

is an intrinsic quantization of the frequency resources given by the subchannel

bandwidth. In our implementation, we let the system evolve according to (3.104)

until successive differences in allocation become smaller than the bandwidth of a

166


0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.27: Example of resource allocation by swarming.

frequency subchannel. At that point, the evolution stops and every SU is allowed

to transmit over the selected channel. The channel chosen by each cognitive node

at the end of the iterations is indicated by the symbol assigned to each vertex

in Fig. 3.27. Each symbol identifies a frequency subchannel uniquely. In our

simulation, the low interference band on the spectrum is divided in 8 different

channels. Hence, the number of available resources is much smaller than the

number of users. From Fig. 3.27 we can observe that the nodes that have picked

up the same channel are never neighbor of each other. This shows the capability

of the proposed method to implement a decentralized mechanism for spatial reuse

of frequencies.

167

3.9 Conclusion

3.9 Conclusion

In this chapter we have proposed a distributed resource allocation strategy for

the access of opportunistic users in cognitive networks based on a swarm model

mimicking the foraging behavior of a swarm, where the interference distribution

over the radio resource domain plays the role of spatial distribution of food. The

swarm mechanism includes an attraction force, useful to minimize the spread over

the resource domain, and a repulsion force, useful to avoid collisions among radio

nodes. In the proposed model, each agent is supposed to listen only to nearby

nodes, in a narrow band spectral interval, over consecutive time slots. This is

useful to limit the complexity of the secondary users’ equipments and, interest-

ingly enough, it produces, as a by-product, an intrinsic capability of the system

to provide spatial reuse of frequency, through a purely decentralized mechanism.

We have derived closed form expressions, validated by numerical results, for the

upper and lower bounds of the final spread as a function of the main system

parameters and of the network topology.

Specific attention has been devoted to the analysis of the proposed swarm-

ing mechanism in the frequency domain, where a local stability analysis has

shown that the introduction of the attraction and unbounded repulsion terms

in the functional to be minimized does not affect the stability of the system.

Furthermore, we have introduced fast swarming methods, useful to improve the

convergence rate and the algorithm’s reaction time in dynamic environments.

Simulation results confirm that these techniques outperform the basic swarming

algorithm. The application of the proposed model to the distributed resource al-

location problem on a time-frequency plane has been also described. We consider

both a static interference scenario, where the interference activity is assumed to

be known and constant along the duration of the swarming algorithm, and a

dynamic interference scenario. In this latter case, the interference activity over

each frequency subchannel is modeled as a continuous-time Markov chain. Inter-

estingly, simulation results show how the swarm tends to stay as close as possible

168

3.9 Conclusion

to time 0, where the prediction of the expected interference is better, under the

constraint of avoiding collisions.

In the last part of the chapter, we have extended the swarming algorithm

for allocating resources in cognitive radio networks considering the presence of

channel imperfections, such as link failures, estimation errors, and quantization

noise. We showed that the swarm converges almost surely to an equilibrium

configuration dependent on the mean graph of the network. We also established

that the resource allocation algorithm is robust against channel imperfections

such as random link failures, quantization noise and estimation errors, whose

effect is only to slow down the convergence process. In particular, reducing the

probability to establish a communication link, the network requires more time

to reach the final equilibrium state. Simulation results show that the cohesion

force introduces cooperation gain among the SU’s and represents an intrinsic

robustness factor of the algorithm. Finally, an adaptive swarming approach was

applied to allocate resources in the frequency domain. The proposed algorithm

adapts the speed of the swarm individuals according to the perceived interference

distribution resulting in an improved allocation performance, convergence speed,

adaptation and learning capability.

169

Chapter 4

Distributed Cooperative

Spectrum Sensing Based on

Diffusion Adaptation

In this chapter, a distributed technique for cooperative spectrum estimation

in cognitive radio systems is introduced, based on a basis expansion model of the

power spectral density (PSD) map in frequency. Joint estimation of the model

parameters enables identification of the (un)used frequency bands, thus facilitat-

ing spatial frequency reuse. The proposed method, based on diffusion adaptation

algorithms [143, 145], estimates and learns the interference profile through local

cooperation and without the need for a central processor. Compared to well stud-

ied incremental methods, diffusion methods do not require the use of a cyclic path

over the nodes and are robust to node and link failures. The diffusion mechanism

endows the networks with powerful learning and tracking abilities that enable the

individual nodes to continue learning even when the cost function changes with

time, exploiting both the time- and spatial-diversity of the data. The diffusion of

information across the network results in various forms of self-organizing behavior

and collective intelligence, which is well-suited to model, e.g., animal swarming.

171

4.1 Introduction

4.1 Introduction

Spectrum sensing is a critical prerequisite in envisioned applications of wire-

less cognitive radio (CR) networks which promise to resolve the perceived band-

width scarcity versus under-utilization dilemma. Creating an interference map

of the operational region plays an instrumental role in enabling spatial frequency

reuse and allowing for dynamic spectrum allocation in a hierarchical access model

comprising primary and secondary users [93], [94]. The non-coherent energy de-

tector has been widely used to this end because it is simple and obviates the need

for synchronization with unknown transmitted signals; see e.g., [157], [158], [159],

and [104]. Power information (or other statistics [155], [156]) collected locally per

CR is fused centrally by an access point in order to decide absence or presence of

a primary user per frequency band. At the expense of commensurate communi-

cation overhead [158], these cooperative sensing and detection schemes have been

shown to increase reliability, reduce the average detection time, cope with fading

propagation effects, and improve throughput [156], [157], [159], [104]. Recently,

the possibility of spatial reuse has received growing attention. It was noticed

that even if a frequency band is occupied, there could be locations where the

transmitted power is low enough so that these frequencies can be reused with-

out suffering from or causing harmful interference to the primary system. These

opportunities are discussed in [160], and a statistical model for the transmitters’

spatial distribution is advocated in [161].

Communications over radio networks is concerned with efficient techniques

for dynamic access to spectral resources [2, 3] and for self-organization (SO) ca-

pabilities. SO is important in femtocell networks, where the deployment of a

potentially large number of user-operated femto-access points makes centralized

schemes hard to implement and prone to heavy signaling traffic. While decentral-

ized resource allocation strategies are certainly more appealing, relying on pure

decentralization with adaptation and learning abilities can lead to inefficient im-

plementations. A more viable approach consists in endowing the radio nodes

172

4.1 Introduction

with the capability to learn from the environment and to exchange information

only with immediate neighbors in order to identify the most appropriate radio

resources. It is only natural that the deployment of cognitive terminals can bene-

fit from highly decentralized radio access and sensing strategies. The deployment

of distributed sensing strategies was proposed, for example, in [172, 173], where

cooperative spectrum sensing techniques exploited the intrinsic sparsity of the

radio resource allocation.

The decentralized approach that is pursued in this work is inspired by behavior

encountered in nature and is meant to endow the cognitive nodes with adaptation

and learning abilities. Biological networks tend to exhibit robust behavior and

are are capable of solving difficult organizational tasks through local cooperation

among the individual agents without central control. In this chapter, we propose

a distributed technique for cooperative spectrum estimation in cognitive radio

systems based on diffusion adaptation algorithms. In comparison with other

distributed approaches that rely on, for example, consensus-based techniques

[132], [133], [134], [136], adaptive networks avoid the need to iterate over data

and do not require all nodes to converge to the same equilibrium (or consensus).

The basic contributions of this chapter are:

1) the exploitation of sparsity in the distributed estimation problem over adaptive

networks;

2) the derivation of mean-square analysis for the sparse diffusion adaptive filter;

3) a real-time distributed spectrum estimation technique based on diffusion adap-

tation techniques;

4) the derivation of the mean-square properties of the diffusion adaptive filter

applied to the spectrum estimation problem.

The chapter is organized as follows. In section 4.2 we describe the basic diffu-

sion algorithm from [145]. Section 4.3 introduces novel regularized diffusion LMS

173

4.2 Diffusion Adaptation

techniques for distributed estimation over adaptive networks, which are able to

exploit sparsity in the underlying system model. Convergence and mean square

analysis of the sparse adaptive diffusion filter show under what conditions we

have dominance of the proposed method with respect to its unregularized coun-

terpart in terms of steady-state performance. Simulation results also confirm the

potential benefits of the proposed filter under the sparsity assumption on the

true coefficient vector. From section 4.4 we illustrate the proposed distributed

spectrum estimation technique based on diffusion adaptation. We first intro-

duce a basis expansion model, which is useful to model the PU’s transmission,

allowing distributed cooperative sensing. Then, we propose a normalized ver-

sion of the Adapt then Combine (ATC) diffusion algorithm [145], which enables

the network to learn and track the time-varying interference profile. Convergence

and mean-square performance analysis of the proposed normalized ATC diffusion

filter, applied to the spectrum estimation problem, is also derived.


In this section we briefly recall some results on adaptive estimation over net-

works from [145]. We consider the problem of distributed estimation, where a

set of nodes is required to collectively estimate some parameter of interest from

noisy measurements by relying solely on in-network processing. Thus, consider a

set of M nodes distributed over some geographic region. At every time instant

k, every node i takes a scalar measurement di(k) of some random process δi(k)

and a 1×N regression vector, ui,k, of some random process υi,k. The objective

is for every node in the network to use the collected data di(k),ui,k to estimate

some parameter vector wo in a distributed manner. In the centralized solution

to the problem, every node in the network transmits its data di(k),ui,k to a

central fusion center for processing. This approach has the disadvantage of being

non-robust to failure by the fusion center. Moreover, in the context of wireless

sensor networks, centralizing all measurements in a single node lacks scalability,

174


and may require large amounts of energy and communication resources [1]. On

the other hand, in distributed implementations, every node in the network com-

municates with a subset of the nodes, and processing is distributed among all

nodes in the network. For the following derivations, we assume the presence of a

linear measurement model where, at every time instant k, every node i takes a

measurement according to the model:

di(k) = ui,kwo + vi(k) (4.1)

where vi(k) is a zero mean random variable with variance σ2v,k, independent of

ui,k for all k and i, and independent of vj(l) for l 6= k and i 6= j. Linear models

as in (4.1) are customary in adaptive filtering since they are able to capture

many cases of interest. The objective of the network is to estimate wo in a fully

distributed manner and in real-time, where each node is allowed to interact only

with its neighbors. That is, the nodes would like to estimate the global parameter

wo that minimizes the following cost function:

Jw(w) =M∑

k=1

E|di(k)− ui,kw|2 (4.2)

where E(·) denotes the expectation operator. Useful diffusive adaptation schemes

were developed for such purpose in [143,145] and their mean-square performance

was studied there in great detail. One example is the so-called Adapt-then-

Combine (ATC) diffusion algorithm in [145], which operates as follows:

ψi,k = wi,k−1 + µi∑

j∈Nicj,iu

∗j,k[dj(k) − uj,kwi,k−1]

(adaptation step)

wi,k =∑

j∈Niaj,iψj,k (diffusion step)

(4.3)

i = 1, . . . ,M , where µi is a positive step-size chosen by node i, and the operator ∗

denotes complex conjugate transposition. The first step in (4.3) is an adaptation

step, where the coefficients cj,i determine which nodes j ∈ Ni should share their

measurements dj(k),uj,k with node i. The second step is a diffusion step where

175

4.3 Sparse Diffusion Adaptation

the intermediate estimates ψj,k, from the neighborhood j ∈ Ni, are combined

through the coefficients aj,i. The non-negative combination matrices C =

cj,i ∈ RM×M and A = aj,i ∈ R

M×M satisfy

cj,i > 0, aj,i > 0 if j ∈ Ni, (4.4)

1TC = 1T , C1 = 1, and 1TA = 1T . (4.5)

At the same way, reversing the order of the processing steps in (4.3), we also

consider the Combine-then-Adapt strategy for the distributed minimization of

(4.44):

ψi,k−1 =∑

j∈Niaj,iwj,k−1 (diffusion step)

(adaptation step)

wi,k = ψi,k−1 + µi∑

j∈Nicj,iu

∗j,k[dj(k)− uj,kψi,k−1]

(4.6)

A detailed convergence and mean-square analysis of these adaptive diffusion

schemes can be found in [145]. In the next section we introduce regularized dif-

fusion LMS techniques for distributed estimation over adaptive networks, which

are able to exploit sparsity in the underlying system model. Convergence and

mean square analysis of the sparse adaptive diffusion filter are also derived and

simulation results confirm the potential benefits of the proposed method under

the sparsity assumption on the true coefficient vector.


For adaptive distributed estimation purposes, several diffusion adaptation

techniques were proposed and studied in [143, 145], where the nodes exchange

information locally and cooperate in order to estimate wo without the need for

a central processor. In many scenarios, the vector parameter wo can be sparse,

containing only a few large coefficients among many negligible ones. Exploiting

the prior information about the sparsity of wo can help improve the estima-

tion performance and this fact has already been investigated in the literature

176


for several years. Specifically, motivated by LASSO [162] and works on com-

pressive sensing [163,164], several algorithms have been proposed in the context

of sparse adaptive filtering frameworks such as LMS [166], RLS [167, 168], and

projection-based methods [169]. A distributed algorithm implementing LASSO

over an ad-hoc network of nodes has been also proposed in the context of spec-

trum sensing for cognitive radio [172] and for sparse linear regression [171]. The

basic idea of these techniques is to introduce a convex penalty, i.e., an ℓ1-norm

term, into the cost function to favor sparsity. However, none of the earlier works

considered the design of adaptive distributed solutions that are able to process

data online and exploit sparsity at the same time. Doing so would endow net-

works with learning abilities and would allow them, for example, to learn the

sparse structure from incoming data recursively and also to track variations in

the sparsity of the underlying vector. In this work, we consider adaptive networks

running diffusion techniques under general constraints enforcing sparsity. In par-

ticular, we consider two convex regularization functions. First, we consider the

ℓ1-norm, which acts as a uniform zero-attractor. Then, to improve the estima-

tion performance, we employ a reweighted regularization to selectively promote

sparsity on the zero elements of w0, rather than uniformly on all the elements.

We provide convergence analysis of the proposed methods, giving a closed form

expression for the bias on the estimate due to the regularization. We also provide

a mean-square analysis, showing the conditions under which the sparse diffusion

filter outperforms its unregularized version in terms of steady-state performance.

Interestingly, it turns out that, if the system model is sufficiently sparse, it is pos-

sible to tune a single parameter to achieve better performance than the standard

diffusion algorithm.

4.3.1 Sparse ATC Diffusion

Assuming the presence of a linear observation model as in (4.1), the cooper-

ative sparse estimation problem can be cast as the distributed minimization of

177


the following cost function:

Jw(w) =

M∑

k=1

E|di(k)− ui,kw|2 + ρf(w) (4.7)

where E(·) denotes the expectation operator, and f(w) is a convex regularization

term weighted by the parameter ρ > 0, which is used to enforce sparsity. Pro-

ceeding as in [145], it is possible to develop several diffusion adaptation schemes

for such purpose. In this paper, we consider the Adapt-then-Combine (ATC)

strategy and refer to the following algorithm as the ATC-sparse diffusion (or

ATC-SD) version:

ψi,k = wi,k−1 + µi∑

j∈Nicj,iu

∗j,k[dj(k) − uj,kwi,k−1]

−µiρ∂f(wi,k−1) (adaptation step)

wi,k =∑

j∈Niaj,iψj,k (diffusion step)

(4.8)

i = 1, . . . ,M , where µi is a positive step-size chosen by node i, the operator ∗

denotes complex conjugate transposition, and ∂f(w) is the sub-gradient of the

convex function f(w). The first step in (4.8) is an adaptation step, where the

coefficients cj,i determine which nodes j ∈ Ni should share their measurements

dj(k),uj,k with node i. The second step is a diffusion step where the interme-

diate estimates ψj,k, from the neighborhood j ∈ Ni, are combined through the

coefficients aj,i. The non-negative combination matrices C = cj,i ∈ RM×M

and A = aj,i ∈ RM×M satisfy (4.4).

In this paper we consider two different convex regularization terms. Motivated

by LASSO [162] and work on compressive sensing [163], we first propose the ℓ1-

norm as penalty function, i.e.,

f1(w) = ‖w‖1, (4.9)

in the global cost function (4.44). This choice leads to an algorithm update in

(4.3) where the subgradient vector is given by ∂f1(w) = sign(w), where sign(x)

178


is a component-wise function defined as

sign(x) =

x/|x|, x 6= 0

0, x = 0. (4.10)

This update leads to what we shall refer to as the zero-attracting (ZA) ATC

diffusion algorithm. The ZA update uniformly shrinks all components of the

vector, and does not distinguish between zero and non-zero elements. Since all

the elements are forced toward zero uniformly, the performance would deteriorate

for systems that are not sufficiently sparse. Motivated by the idea of reweighting

in compressive sampling [164], we also consider the following cost function:

f2(w) =N∑

n=1

log(1 + ε|wn|), (4.11)

which behaves more similarly to the l0-norm than the l1-norm [164], thus enhanc-

ing the sparsity recovery of the algorithm. The algorithm in (4.3) is then updated

by using

∂f2(w) = εsign(w)

1 + ε|w|, (4.12)

leading to what we shall refer to as the reweighted zero-attracting (RZA) ATC

diffusion algorithm. The update in (4.12) selectively shrinks only the components

whose magnitudes are comparable to 1/ε, and there is little effect on components

satisfying |wn| ≫ 1/ε.

4.3.2 Performance Analysis

In this section we analyze the performance of the ATC-SD algorithm. In what

follows we view the estimates wi,k as realizations of a random process ωi,k and

analyze the performance of the algorithm in terms of its mean square behavior.

To proceed with the analysis, we assume a linear measurement model as in

(4.1). Using (4.8), we define the error quantities wk,i = wo − wi,k, ψi,k =

179


wo −ψi,k, and the global vectors:

wk =

w1,k

...

wM,k

, wk =

w1,k

...

wM,k

, ψi =

ψ1,k...

ψM,k

. (4.13)

We also introduce a diagonal matrix

M = diagµ1IN , . . . , µMIN, (4.14)

and the extended weighting matrices

C = C ⊗ IN , A = A⊗ IN , (4.15)

where ⊗ denotes the Kronecker product operation. We further introduce the

following random quantities:

Dk = diag

M∑

j=1

cj,1u∗j,kuj,k, . . . ,

M∑

j=1

cj,Mu∗j,kuj,k

(4.16)

gk = CTcolu∗

1,kv1(k), . . . ,u∗M,kvM (k). (4.17)

Then, we can write (4.3) in compact form as:

ψk = wk−1 − M [Dkwk−1 + gk] + ρM∂f(wk−1)

wk = ATψk (4.18)

where ∂f(wk−1) = col[∂f(w1,k−1), . . . ,∂f(wM,k−1)], or, equivalently,

wk = AT[I − MDk]wk−1 − A

TMgk + ρA

TM∂f(wk−1). (4.19)

Mean Stability

Assuming all regressors ui,k are spatially and temporally independent and

taking the expectation of (4.19), we get

Ewk = AT [I − MEDk]Ewk−1 + ρATME∂f(wk−1). (4.20)

180


Since the subgradient vector ∂f(wk−1) has bounded entries, the algorithm (4.56)

converges in the mean if the matrix AT[I−MEDk] is a stable matrix. Since the

entries on the columns of AT

add up to one, and since M is diagonal, we can

show that the previous condition holds if the matrix I −MD, where D = EDk,

is stable. Using (4.51) we conclude that the algorithm converges in the mean for

any step-size satisfying:

0 < µi <2

λmax

(

∑Mj=1 cj,iRu,j

) , i = 1, . . . ,M, (4.21)

where λmax(X) denotes the maximum eigenvalue of a Hermitian matrix X, and

Ru,i = Eu∗i,kui,k is the covariance matrix of the regression data at the i-th node.

Furthermore, taking the limit as k →∞ of equation (4.20), we have

Ew∞ = wo − ρ[

I − AT [I − MD

]

]−1A

TME∂f(w∞), (4.22)

thus concluding that the estimate wk is asymptotically biased; moreover, the

smaller the value of ρ, the smaller the bias.

Mean-Square Performance

In this section we examine the mean-square performance of the diffusion filter

(4.56). Now, following the energy conservation framework of [143, 145], we can

evaluate the weighted norm of wk, obtaining:

E‖wk‖2Σ= E‖wk−1‖

2Σ

′ + E[g∗kMAΣATMgk] + φΣ,k(ρ) (4.23)

where Σ is an Hermitian positive-definite matrix that we are free to choose, and

Σ′ = E(I −DkM )T AΣAT(I − MDk), (4.24)

φΣ,k(ρ) = ρβΣ,k

(

ρ−αΣ,k

βΣ,k

)

, (4.25)

where

βΣ,k = E‖∂f(wk−1)‖2MAΣAT M

> 0, (4.26)

αΣ,k = −2E∂f(wk−1)TMAΣA

T [I − MD

]

wk−1. (4.27)

181


Moreover, setting

G = E[gkg∗k] = C

Tdiagσ2v,1Ru,1, . . . , σ

2v,MRu,MC, (4.28)

we can rewrite (4.64) in the form

E‖wk‖2Σ = E‖wk−1‖

2Σ

′ +Tr[ΣATMGMA] + φΣ,k(ρ) (4.29)

where Tr(·) denotes the trace operator. Let σ = vec(Σ) denote the vector that is

obtained by stacking the columns of Σ on top of each other. Using the Kronecker

product property vec(UΣV ) = (V T ⊗ U)vec(Σ), we can vectorize Σ′ in (4.67)

as σ′ = vec(Σ′) = Fσ, where the matrix F is given by:

F = (I ⊗ I)I − I ⊗ (DM)− (DM )⊗ I

+ E(DkM)⊗ (DkM)(A⊗ A). (4.30)

Then, using the property Tr(ΣX) = vec(XT )Tσ and taking the limit as k →∞

(assuming the step-sizes are small enough to ensure convergence to steady-state),

we deduce from (4.29) that:

E‖w∞‖2Σ−Σ

′ = [vec(ATMGMA)]Tσ + ρβΣ,∞

(

ρ−αΣ,∞

βΣ,∞

)

.

The steady-state mean-square deviation (MSD) of the network is defined as:

MSDnet = limk→∞

1

M

M∑

i=1

E‖wi,k‖2. (4.31)

Then, if the step sizes µi are small enough so that the matrix (I − F ) is

invertible and choosing σ = (I − F )−1vec(I ⊗ I), the network MSD tends to

MSDnet =1

M[vec(A

TMGTMA)]T (I − F )−1vec(I ⊗ I)

+1

MρβΣ,∞

(

ρ−αΣ,∞

βΣ,∞

)

. (4.32)

182


The first term on the right-hand side of (4.75) is the network MSD of the standard

diffusion algorithm (compare with (48) in [145]), whereas the second term is due

to the regularization. Then, if

αΣ,∞ > 0, and 0 < ρ <αΣ,∞

βΣ,∞, (4.33)

the ATC-SD algorithm would perform better than the standard diffusion [145].

Let us examine the interpretation of the condition ασ,∞ > 0, where ασ,i is

given by (4.27), relating this condition to the sparsity of the vector wo. In-

deed, since f(·) is a convex regularization function, it holds that f(x + y) −

f(x) ≥ ∂f(x)Ty. Then, choosing x = w∞ and y = BΣ(wo − w∞), where

BΣ = 2MAΣAT [I − MD

]

, the first condition in (4.33) can be recast as

αΣ,∞ ≥ E[f(w∞)− f(w∞ +Bσ(wo −w∞))] > 0. (4.34)

If the step-sizes are sufficiently small, we can approximate BΣ ⋍ 2MAΣAT,

neglecting the second term that depends on µ2. Then, we have wB∞ = w∞ +

BΣ(wo−w∞) ⋍ w∞−2MAΣA

T(w∞−w

o). This expression can be interpreted

as a gradient descent update minimizing the function ‖w∞−wo‖2

AΣAT , yielding

wB∞ closer to wo than w∞. As a consequence, if wo is sparse, wB

∞ will be more

sparse than w∞ and the condition in (4.34) will likely be true. Then, by selecting

properly the sparsity coefficient ρ, the ATC-SD algorithm will have better MSD

than the standard ATC diffusion algorithm. On the other hand, if wo is not

sparse, condition (4.34) in general would not be true, thus leading the ATC-SD

algorithm to perform worse than standard ATC diffusion.

4.3.3 Numerical Results

In this section, we provide some numerical examples to illustrate the perfor-

mance of the ATC-SD algorithm. We consider a connected network composed

of 20 nodes. The regressors have size N = 50 and are zero-mean white Gaus-

sian distributed with covariance matrices Ru,i = σ2uI, with σ2u = 0.1, for all

183


0 200 400 600 800 1000 1200 1400 1600 1800

−30

−25

−20

−15

−10

−5

0

5

10

15

Iteration index

MS

D (

dB

)LMS

ZA LMS

RZA LMS

ATC Diffusion

ZA ATC Diffusion

RZA ATC Diffusion

Figure 4.1: Transient network MSD for the non-cooperative approaches LMS

[190], ZA-LMS [166], RZA-LMS [166], and the diffusion techniques ATC [145],

ZA-ATC (eq.(4.3)-(4.9)), RZA-ATC (eq.(4.3)-(4.11)).

i. The background white noise power is set to σ2v = 0.01. The first example

aims to show the tracking and steady-state performance for the ATC-SD algo-

rithm. In Fig. 4.1, we report the learning curves in terms of network MSD of

6 different adaptive filters: ATC diffusion LMS [145], ZA-ATC (eq.(4.3)-(4.9))

and RZA-ATC diffusion (eq.(4.3)-(4.11)), and the corresponding non cooperative

approaches from [166]. The simulations use a value of µ = 0.2 and the results

are averaged over 100 independent experiments. The sparsity parameters are set

equal to ρLMS = 5 × 10−3 for the non cooperative approaches, ρZA = 10−3 for

ZA-ATC, ρRZA = 0.25 × 10−3 for RZA-ATC, and ǫ = 10. In this simulation, we

consider diffusion algorithms without measurement exchange, i.e., C = I, and a

combination matrixA that simply averages the estimates from the neighborhood,

hence, such that aj,i = 1/Ni for all j. Initially, only one of the 50 elements of

wo is set equal to one while the others are equal to zero, making the system very

sparse. After 600 iterations, 25 elements are randomly selected and set equal to

184


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007−4

−3

−2

−1

0

1

2

Sparsity parameter ρ

Diffe

rential M

SD

(d

B)

Sparsity ratio = 1/50





Figure 4.2: Differential MSD versus sparsity parameter ρ for ZA-ATC Diffusion

LMS, for different degrees of system sparsity.

1, making the system have a sparsity ratio of 25/50. After 1200 iterations, all the

elements are set equal to 1, leaving a completely non-sparse system. As we see

from Fig. 5.4, when the system is very sparse both ZA-ATC and RZA-ATC yield

better steady-state performance than standard diffusion. The RZA-ATC outper-

forms ZA-ATC thanks to the reweighted regularization. When the vector wo is

only half sparse, the performance of ZA-ATC deteriorates, performing worse than

standard diffusion, while RZA-ATC has the best performance among the three

diffusion filters. When the system is completely non-sparse, the RZA-ATC still

performs comparably to the standard diffusion filter. Finally, we can also notice

the gain of the diffusion schemes with respect to the non-cooperative approaches

from [166,190]. To quantify the effect of the sparsity parameter ρ on the perfor-

mance of the ATC-SD filters, we consider two additional examples. Considering

the same settings of the previous simulation, in Fig. 4.2, we show the behavior

of the difference (in dB) between the network MSD of ATC-ZA and standard

diffusion, versus ρ, for different sparsity degrees of wo. The results are averaged

185


0 0.001 0.002 0.003 0.004 0.005 0.006−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

Sparsity parameter ρ

Diffe

rential M

SD

(d

B)







Figure 4.3: Differential MSD versus sparsity parameter ρ for RZA-ATC Diffusion

LMS, for different degrees of system sparsity.

over 100 independent experiments and over 100 samples after convergence. As

we can see from Fig. 4.2, reducing the sparsity of wo, the interval of ρ values

that yield a gain for ATC-ZA with respect to standard diffusion becomes smaller,

until it reduces to zero when the system is not sparse enough. In this case, in

fact, the ATC-ZA diffusion algorithm only introduces a bias in the final estimates

without improving the system performance. In Fig. 4.3, we repeat the same ex-

periment considering the ATC-RZA algorithm. As wee can see, ATC-RZA gives

better performance than ZA-ATC and yields a performance loss with respect to

standard diffusion, for any ρ, only when the vector w0 is completely non-sparse.

This is an effect of the selective shrinking of the algorithm, which acts only on

the components whose magnitudes are comparable to 1/ε, whereas there is little

effect on components satisfying |wn| ≫ 1/ε.

In the next section, we introduce a basis expansion model useful to describe

PU transmissions in a cognitive radio system. This model will be instrumental

to enable the network with adaptive spectrum estimation capabilities.

186

4.4 Basis Expansion Model of the Spectrum


Consider Ntx active PU’s transmitters and let yi(k) denote the signal received

at time k by the SU i, given by the superposition of the transmitted signals

uq(k), q = 1, . . . , Ntx convolved with a linear, and possibly time-varying, fading

tapped-delay-line channel with impulse response hiq(k, l)Liq

l=1, and observed in

the presence of additive white noise n(k):

yk(i) =

Ntx∑

q=1

Liq∑

l=1

hiq(k, l)uq(k − l) + n(k). (4.35)

We consider the two following assumptions on the transmitted signals and on the

communication channels.

Assumption 1 : The source signals uq(k) are stationary, mutually uncorrelated

and independent of the channels hiq(k, l).

Assumption 2 : The channels hiq(k, l) are zero mean and uncorrelated across

the lag variable l and the spatial variables q and i.

Spatial uncorrelatedness of the channels is well justified since PU’s are physically

sufficiently far apart relative to the high carrier frequencies, which determine

short wavelengths. Since the channels are uncorrelated across lags l, the channel

gain piq(k) is frequency invariant, but possibly time-variant, and given by

piq(k) =

Lqi∑

l=1

σ2iq(k, l), (4.36)

where σ2iq(k, l) = E|hiq(k, l)|2. This model is quite general and takes into account

the possibility of node mobility and possible variations in the channel conditions

over time. In this paper, we assume the gain piq(k) can be acquired via train-

ing. Then, cooperation between primary and secondary users is allowed. The

case where such cooperation is not allowed was analyzed in [172], where the au-

thors used a grid of candidate PU’s locations and a known model to describe

187


15 20 25 300

0.5

1

1.5

Frequency (MHz)

PS

D (m

W/H

z)

Figure 4.4: Example of basis expansion using Gaussian pulses. The dotted curves

represent the Gaussian basis functions, whereas the continuous curve denotes the

behavior of a generic interference profile described by 6 Gaussian pulses.

the path loss between transmitter and receiver. In the following, we introduce a

basis expansion model, which is useful to model the PU’s transmission, allowing

distributed cooperative sensing.

Let Φq(f) denote the power spectral density (PSD) of the stationary random

signal uq(k) transmitted by the q-th primary user. The PSD can be represented

as a linear combination of some preset J basis functions, say, as:

Φq(f) =J∑

j=1

bj(f)wqj = bT0 (f)wq (4.37)

where b0(f) = [b1(f), ..., bJ (f)]T is the vector of basis functions evaluated at

frequency f , wq = [wq1, ..., wqJ ] is a vector of weighting coefficients representing

the power transmitted by the q-th PU over each basis, and J is the number

of basis functions. For J sufficiently large, the basis expansion in (4.37) can

approximate well the transmitted spectrum. Several choices for the set of basis

bj(f)Jj=1 are possible. In particular, we consider continuously differentiable

188


basis functions, such as raised cosines, Gaussian pulses, etc. An example of basis

expansion using Gaussian pulses is shown in Fig. 4.4. Assuming Ntx active users

are transmitting, the overall transmitted spectrum can be expressed as:

IT (f) =

Ntx∑

q=1

J∑

j=1

bj(f)wqj = bT1 (f)w

o (4.38)

where wo = [w1, . . . ,wNtx] ∈ RJNtx and b1(f) = 1⊗b0(f), with ⊗ and 1 ∈ R

Ntx

denoting, respectively, the Kronecker product operation and the vector of all ones.

The propagation medium introduces path loss attenuation between primary and

secondary users. Let piq(k) be the path loss coefficient between the q-th primary

user transmitter and the i-th secondary user. Then, considering a single source q,

due to the assumptions 1 and 2, we can express the autocorrelation of the signal

received by SU i in absence of noise, as

φi(m) = φq(m)

Liq∑

l=1

E|hiq(k, l)|2, (4.39)

and hence the received PSD, at time k, as Ii(f, k) = piq(k)Φq(f). Considering

now the presence of Ntx active PUs and receiver noise, under the assumption of

spatial uncorrelatedness of the channels and sources, the signal received by the

secondary node i can be expressed as:

Ii(f, k) =

Ntx∑

q=1

piq(k)

J∑

j=1

bj(f)wqj + σ2n,i = bTi,k(f)w

o + σ2n,i, (4.40)

where pk,i = [pk1(i), ..., pkNtx(i)] ∈ RNtx is the vector of path-loss coefficients

between every transmitter and the i-th receiver, σ2n,i is the noise power at the

i-th receiver node, and

bi,k(f) = pi,k ⊗ b0(f). (4.41)

Expression (4.40) models the power received by node i in terms of an unknown

vector wo; this vector represents the expansion of the received power in the basis

defined by the bi,k(f).

189

4.5 ATC Diffusion for Adaptive Spectrum Estimation


At every time instant k, every node i observes noisy measurements of the

PSD Ii(f, k) described by (4.40) over Nc frequency samples fm = fmin : (fmax −

fmin)/Nc : fmax, for m = 1, . . . , Nc, according to the model:

dmi (k) = bTi,k(fm)wo + σ2n,i + vmi (k) (4.42)

where wo is the true vector parameter, and vmi (k) is a zero mean random variable

with variance σ2v,m. The temporal index k in the regressor expression bTi,k(fm)

takes into account the possibility of node mobility and possible variations in the

channel conditions over time. The receiver noise power σ2n,i can be pre-estimated

with high accuracy using an energy detection over an idle band. It can then

be removed from expression (4.1). Collecting measurements over Nc contiguous

channels, we obtain a vector linear model:

di,k = Bi,kwo + vi,k (4.43)

where Bi,k = [bTi,k(fm)]Ncm=1 ∈ R

Nc×JNtx and vi,k is a zero mean random vector

with covariance matrix Rv,i. Given the interference measurements di,k across

all M secondary users, these users can now cooperate to estimate the modeling

vector wo in a distributed and adaptive manner. They can do so by seeking to

minimize the the following cost function:

JE(w) =M∑

i=1

E‖di,k −Bi,kw‖2 (4.44)

where E(·) denotes the expectation operator. The minimization of (4.44) can be

computed using a centralized algorithm, which can be run by a fusion center once

all nodes transmit their data di,k,Bi,k, for all k, to it. However, our emphasis

is on a distributed solution, where the nodes estimate the interference profile

by relying solely on in-network processing within their neighborhoods. Several

diffusion adaptation schemes have been developed for such purpose in [143,145].

190


In this paper, we employ a vector version of the Adapt-then-Combine (ATC) al-

gorithm without measurement exchange from [145]. For the vector minimization

problem in (4.44), the ATC algorithm reads as follows:

ψi,k = wi,k−1 + µiH i,kBTi,k[di,k −Bi,kwi,k−1] (adaptation step)

wi,k =∑

j∈Nicijψj,k (combination step)

(4.45)

where µi is a positive step-size chosen by node i, and Hi,k ∈ RJNtx×JNtx can be

properly chosen to normalize the algorithm in (5.6). The first step in (5.6) involves

local adaptation where node k updates using the new observations di,k,Bi,k.

The choiceH i,k = I, for all i, leads to the classical ATC diffusion scheme, whereas

by selecting Hi,k = (BTi,kBi,k)

†, where (·)† denotes the pseudo-inverse operation,

leads to an approximate Newton version with improved convergence speed. The

second step in (5.6) is a combination step where the intermediate estimates ψj,k,

from the neighborhood j ∈ Ni, are combined through the coefficients cij. The

combination matrix C = cij ∈ RM×M satisfies cij ≥ 0 if j ∈ Ni and 1TC = 1T .

The resulting estimate of node i at time k is denoted by wi,k. In the case in which

the unknown parameter wo varies slowly with time, the ATC diffusion algorithm

allows online tracking of the interference profile variations.

4.5.1 Performance Analysis

In this section we analyze the performance of the normalized diffusion algo-

rithm (5.6) following the approach of [145] by extending it to the case of a vector

linear model as in (4.43). In what follows we view the estimates wi,k as realiza-

tions of a random process ωk,i and analyze the performance of the algorithms in

terms of their mean square behavior.

We consider a general algorithmic form that includes various normalized diffu-

sion algorithms as special cases. Thus, we consider a general normalized diffusion

191


filter of the form:

φi,i =

M∑

j=1

c1,j,iwj,k−1

ψi,k = φi,k + µi

M∑

j=1

sj,iH i,kBTi,k[di(k)−Bi,kφi,k] (4.46)

wi,k =

M∑

j=1

c2,j,iφi,k

i = 1, . . . ,M , where the coefficients c1,j,i,sj,i and c2,j,i are generic non-

negative real coefficients corresponding to the (j, i) entries of the matrices C1,

S, and C2, respectively, satisfying

1TC1 = 1T ,1TS = 1T ,1TC2 = 1T . (4.47)

Different types of algorithms can be obtained as special cases of (4.46) by choosing

different matrices C1,S,C2. The ATC diffusion algorithm without measure-

ment exchange in (5.6) is obtained by choosing C1 = S = I and C2 = C. To

proceed with the analysis, we assume a linear measurement as in (4.43). Us-

ing (4.46), we define the error quantities wi,k = wo − wi,k, ψi,k = wo − ψi,k,

φi,k−1 = wo − φi,k−1 and the global vectors:

wk =

w1,k

...

wM,k

, ψk =

ψ1,k...

ψM,k

, φk−1 =

φ1,k−1...

φM,k−1

. (4.48)

We also introduce the diagonal matrix

M = diagµ1IJNtx , . . . , µMIJNtx, (4.49)

and the extended weighting matrices

C1 = C1 ⊗ IJNtx , C2 = C2 ⊗ IJNtx, S = S ⊗ IJNtx . (4.50)

192


We further introduce the matrices

Dk = diag

M∑

j=1

sj,iHj,kBTj,kBj,k

M

i=1

, (4.51)

gk = ST· col

H i,kBTi,kvi,k

M

i=1. (4.52)

The matrices H i,k and Bi,k depend on the path loss vector pi,k ∈ RNtx of node

i at time k. Let γk =[

pT1,k, . . . ,pTM,k

]T∈ R

MNtx denote the overall path loss

vector of the network at time k. Then, the noise vector gk is a function of γk and

we have

φk−1 = CT1 wk−1 (4.53)

ψk = φk−1 − M [Dk(γk)φk−1 + gk(γk)] (4.54)

wk = CT2 ψk (4.55)

or, equivalently,

wk = CT2 [I − MDk(γk)]C

T1 wk−1 − C

T2 Mgk(γk). (4.56)

Mean Stability

Assuming the regression data are spatially and temporally white and taking

the expectation of (4.56), we get

Ewk = CT2 [I − MEDk(γk)]C

T1 Ewk−1 (4.57)

The algorithm (4.56) converges in the mean if the matrix CT2 [I−MDk(γk)]C

T1 ,

where Dk(γk) = EDk(γk), is a stable matrix for all k. To proceed, we call upon

results from [145], [147], and [148]. Following [147], let z = colzT1 , . . . ,zTM

T

denote a vector that is obtained by stacking M subvectors on top of each other

(as is the case with wk). The block maximum norm of z is defined as

‖z‖b,∞ = max1≤k≤M ‖zk‖, (4.58)

193


where ‖ · ‖ denotes the Euclidean norm of its vector argument. Likewise, the

induced block maximum norm of a block matrix B is defined as

‖B‖b,∞ = maxz 6=0‖Bz‖b,∞‖z‖b,∞

. (4.59)

Following [148] and applying the triangle inequality of norms to (4.57), we find

that

‖Ewk‖b,∞ ≤ ‖CT2 ‖b,∞ · ‖I − MDk(γk)‖b,∞ · ‖C

T2 ‖b,∞ · ‖Ewk−1‖b,∞. (4.60)

It was argued in [147] that ‖CT1 ‖b,∞ and ‖C

T2 ‖b,∞ are bounded by one. Therefore,

for the vector Ewk to converge to zero, it suffices to require

supk‖I − MDk(γk)‖b,∞ < 1. (4.61)

Then, exploiting the inequality in (4.61) and the structure of the matrices in

(4.49) and (4.51), we conclude that wk is asymptotically unbiased if the step-size

satisfy

0 < µi <2

λmax(Dk(γk)), for all i, k, (4.62)

where λmax(X) denotes the maximum eigenvalue of a Hermitian matrix X. If

H i,k in (5.6) is chosen such that H i,k = (BTi,kBi,k)

†, exploiting the expression

in (4.51), we have that the algorithm converges in the mean for any step-size

satisfying

0 < µi < 2, for all i. (4.63)

Mean-Square Performance

In this section we examine the mean-square performance of the adaptive diffu-

sion filter (4.46). Following the energy conservation arguments of [143,145,191],

we evaluate the weighted norm of wk:

E‖wk‖2Σ

= E‖wk−1‖2C1(I−Dk(γk)M)T C2ΣC

T2 (I−MDk(γk))C

T1

+

+ E[gTk (γk)MC2ΣCT2 Mgk(γk)] (4.64)

194


where Σ is an arbitrary Hermitian positive-definite matrix that we are free to

choose. If we let

G(γk) = E[gk(γk)gTk (γk)] = S

TEdiagH i,kB

Ti,kRv,iBi,kH

Ti,k

Mi=1S, (4.65)

then we can rewrite (4.64) as a variance relation of the form

E‖wk‖2Σ= E‖wk−1‖

2Σ

′ +Tr[ΣCT2 MG(γk)MC2] (4.66)

where Tr(·) is the trace operator, and

Σ′ = C1C2ΣCT2 C

T1 − C1MC2ΣC

T2 C

T1 − C1C1ΣC

T2 MC

T1

+ C1MC2ΣCT2 MC

T1 . (4.67)

We further introduce the notation

σ = vec(Σ), and Σ = vec−1(σ), (4.68)

where the vec(·) notation stacks the columns of the matrix Σ on top of each

other, and vec−1(·) is the inverse operation. We will also use the notation ‖z‖2σ

or ‖z‖2Σ

to interchangeably denote the same squared weighted norm of a vector

z. Using the Kronecker product property [191]:

vec(UΣV ) = (V T ⊗U)vec(Σ) (4.69)

and the fact that the expectation and vectorization operators commute, we can

rewrite Σ′ in (4.67) as σ′ = vec(Σ′) = Fσ, where the matrix F is given by

F = (C1 ⊗ C1)I − I ⊗ (Dk(γk)M)− (Dk(γk)M )⊗ I +

+ E(Dk(γk)M)⊗ (Dk(γk)M )(C2 ⊗ C2). (4.70)

Assumption : The path loss vector γk → γ0, where γ0 is a fixed constant

vector, as k →∞.

Then, using the property Tr(ΣX) = vec(XT )Tσ and taking the limit of

(4.66) as k →∞, we can recast (4.66) as follows:

E‖w∞‖2σ = E‖w∞‖

2Fσ

+ [vec(CT2 MG(γ0)

TMC2)]Tσ. (4.71)

195

4.6 Conclusion

The steady-state mean-square deviation (MSD) at node k is defined as:

MSDi = limk→∞

E‖wi,k‖2. (4.72)

Then, if the step sizes µi are small enough so that the matrix (I − F ) is

invertible and choosing σ = (I − F )−1mi, the MSD of node i tends to

MSDi = [vec(CT2 MG(γ0)

TMC2)]T (I − F )−1mi, (4.73)

where mi = vec(diag(ei)⊗ IJNtx), with ei denoting the column vectors with a

unity entry at position i and zeros elsewhere. The network MSD is defined as the

average MSD across all nodes in the network and is given by

MSDnet =1

M

N∑

i=1

MSDi. (4.74)

Then, from (4.71), the network MSD can be obtained as:

MSDnet =1

M[vec(C

T2 MG(γ0)

TMC2)]T (I − F )−1m, (4.75)

where m = vec(IJNtx ⊗ IJNtx).

In the following chapter, we will illustrate how these theoretical expressions

match well with simulation results.

4.6 Conclusion

The key challenge in developing cognitive wireless transceivers is enabling

them to sense the ambient power spectral density at arbitrary locations in space.

In this chapter, we addressed this challenging task through a parsimonious basis

expansion model of the PSD in frequency. This model reduces the sensing task

to estimating a common vector of unknown parameters. The resulting estimator

relies on diffusion adaptation algorithms, where the cognitive radios exchange

information locally only with their one-hop neighbors, eliminating the need for a

fusion center.

196

4.6 Conclusion

First, we have proposed a novel class of diffusion LMS strategies, regularized

by convex sparsifying penalties, for distributed estimation over adaptive net-

works. Convergence and mean square analysis of the sparse adaptive diffusion

filter show under what conditions we have dominance of the proposed method

with respect to its unregularized counterpart in terms of steady-state perfor-

mance. Two different penalty functions have been employed, the ℓ1-norm, which

uniformly attracts to zero all the vector elements, and a reweighted function,

which selectively shrinks only the elements with small magnitude. Numerical re-

sults show the potential benefits of using such strategies. Other penalty functions

can also be useful. Adaptive diffusion strategies for the distributed optimization

of convex cost functions are further considered in [154].

Then, we have illustrated the proposed distributed spectrum estimation tech-

nique based on diffusion adaptation. We introduce a basis expansion model,

which is useful to model the PU’s transmission, allowing distributed cooperative

sensing. Then, we have proposed a normalized version of the Adapt then Com-

bine (ATC) diffusion algorithm, which enables the network to learn and track

the time-varying interference profile. Convergence and mean-square performance

analysis of the proposed normalized ATC diffusion filter, applied to the spectrum

estimation problem, has been derived.

197

Chapter 5

Swarming for Dynamic Radio

Access Based on Diffusion

Adaptation

The goal of this chapter is to study the learning abilities of adaptive networks

in the context of cognitive radio networks and to investigate how well they assist

in allocating power and communications resources in the frequency domain. The

allocation mechanism is based on a social foraging swarm model that lets every

node allocate its resources (power/bits) in the frequency regions where the inter-

ference is at a minimum while avoiding collisions with other nodes. We employ

adaptive diffusion techniques to estimate the interference profile in a cooperative

manner and to guide the motion of the swarm individuals in the resource domain.

The resulting bio-inspired network cooperatively estimates the interference pro-

file in the resource domain of a cognitive network and allocates resources through

purely decentralized mechanisms. The proposed approach endows the cognitive

network with powerful learning and adaptation capabilities, allowing fast reaction

to dynamic changes in the spectrum.

199

5.1 Swarm Model

The basic contributions of this chapter are:

1) the extension of the social foraging model proposed in chapter 3 to incor-

porate a real-time distributed spectrum estimation technique based on diffusion

adaptation;

2) the application of the proposed procedure to the dynamic resource allocation

problem in the frequency domain.

The chapter is organized as follows. In Section 5.1 we describe the swarm model,

formulating the search of available time/frequency slots as the distributed mini-

mization of a time-varying global potential function. Section 5.2 illustrates the

proposed distributed spectrum estimation technique based on diffusion adapta-

tion, which enables the network to learn and track the time varying interference

profile. In section 5.3, combining the diffusion step from section 5.2 and the

swarming behavior from section 5.1, we illustrate how the proposed adaptive

swarming algorithm performs dynamic resource allocation. Section 5.4 provides

some numerical examples aimed to illustrate the theoretical findings and assess

the performance and adaptation capabilities of the proposed technique.

5.1 Swarm Model

In this section, we describe an improved version of the swarm-based resource

allocation strategy proposed in chapter 2, extending the swarming algorithm to

dynamic environments. For this purpose, we denote by Ii(xi, t) ∈ C1 : Rn ×

R → R the interference power over the slot having coordinate vector xi (e.g.,

a frequency subchannel or a time-frequency slot) perceived by node i at time t.

The resource allocation problem can then be formulated mathematically as the

search of the resource vector x, from the whole population of M cognitive nodes,

that minimizes in a distributed fashion the global function:

J(x, t) =

M∑

i=1

Ii(xi, t) +1

2

M∑

i=1

M∑

j=1

aijJar(‖xj − xi‖), (5.1)

200

5.1 Swarm Model

where x := (xT1 , . . . ,x

TM )T , and the second term on the right-hand side of (3.2)

incorporates an attraction/repulsion potential function given by

Jar(‖xj − xi‖) = Ja(‖xj − xi‖)− Jr(‖xj − xi‖). (5.2)

This potential incorporates a short range repulsion term Jr(‖xj − xi‖), whose

effect is to avoid collisions among the cognitive nodes, and a long range attraction

term Ja(‖xj − xi‖), whose goal is to induce a swarm cohesion behavior, e.g. to

avoid an excessive spread of the selected radio resources in the time-frequency

domain. In summary, minimizing (3.2) leads each node to dynamically allocate

its own resources in time-frequency regions where there is less interference, helps

to avoid conflicts among users, and limits the spread of the occupied domain.

Remark : Differently from the swarm model proposed in chapter 3, the function

(5.1) takes into account the time variability of the interference profile Ii(xi, t)

sensed by each node. This is an important difference because it enables the

swarm to dynamically allocate in the resource domain, tracking the changes in

the interfering environment and reacting to them. A similar scenario arises in the

modeling of bird flight formations through adaptive networks [149, 150], where

the total upwash (cost) function evolves dynamically as the birds move in search

of the optimal (peak) location.

Considering again the swarm analogy, the occupied zones in the resource domain

take the role of dangerous regions that must be avoided by the swarm individuals

as fast as possible, while idle bands represent regions rich of food that the agents

have to occupy reducing their speed. Mimicking this natural learning capability,

we consider a distributed minimization of (3.2) based on a scaled gradient descent

optimization, so that every node starts with an initial guess, let us say xi[0], and

then updates its resource allocation vector xi in time according to the following

discrete-time implementation:

xi[k + 1] = xi[k]− αi[k]∇xiJ(x, kT ) (5.3)

201

5.2 Diffusion Adaptation for Cooperative Spectrum Sensing

= xi[k]− αi[k]

∇xiIi(xi, kT )−M∑

j=1

aij g(xj[k]− xi[k])

,

i = 1, . . . ,M , where T is the sampling time, k is the time index, αi[k] > 0 is a

positive iteration-dependent step-size and g(·) is a vector function defined as in

(3.4). The step size is given by

αi[k] = f(Ii(xi, kT )) ∈ [αmin, αmax] > 0, (5.4)

where f(·) is a monotonically increasing function of the interference power per-

ceived at time i by every node at its current position on the resource domain.

Examples include linear, quadratic, logarithmic functions, etc. The goal is to

accelerate the motion of the resources perceiving a high interference and, at the

same time, to slow down the resources that are allocating on idle sub-bands. This

adaptive behavior considerably improves the reaction capability of the algorithm

to changes in the environment. In this chapter, we consider dynamic swarm-

ing in the frequency domain, where xi ∈ R is a scalar denoting the position of

the i-th resource on the frequency axis. To update the swarming behavior in

(5.3), the SUs need to estimate the interference profile Ii(xi, kT ) and its gradi-

ent ∇xiIi(xi, kT ) on the current position xi in the frequency domain, at time k.

Then, in the next section, we show how to adaptively estimate these quantities

in a distributed manner and through local cooperation.

5.2 Diffusion Adaptation for Cooperative Spectrum

Sensing

As shown in the previous chapter, At every time instant k, every node i

observes noisy measurements of the PSD Ii(f, k) described by (4.40) over Nc

frequency samples fm = fmin : (fmax − fmin)/Nc : fmax, for m = 1, . . . , Nc,

according according to the vector linear model:

di,k = Bi,kwo + vi,k (5.5)

202

5.2 Diffusion Adaptation for Cooperative Spectrum Sensing

where Bi,k = [bTi,k(fm)]Ncm=1 ∈ R

Nc×JNtx and vi,k is a zero mean random vector

with covariance matrix Rv,i. The temporal index k in the regressor expression

Bi,k takes into account the possibility of node mobility and possible varia-

tions in the channel conditions over time. Given the interference and regression

measurements di,k,Bi,k across all M secondary users, these users can now co-

operate to estimate the modeling vector wo in a distributed and adaptive manner.

For this purpose, we employ a vector version of the Adapt-then-Combine (ATC)

algorithm without measurement exchange from [145]. For the vector minimiza-

tion problem in (4.44), the ATC algorithm reads as follows:

ψi,k = wi,k−1 + µiH i,kBTi,k[di,k −Bi,kwi,k−1] (adaptation step)

wi,k =∑

j∈Nicijψj,k (combination step)

(5.6)

where µi is a positive step-size chosen by node i, and Hi,k ∈ RJNtx×JNtx can be

properly chosen to normalize the algorithm in (5.6). In the case in which the un-

known parameter wo varies slowly with time, the ATC diffusion algorithm allows

online tracking of the interference profile variations, thus enabling the swarming

agents to dynamically allocate resources on the frequency domain. Using its local

estimates wi,k, and using (4.40), every node i can determine an estimate for the

transmitted interference profile and of its derivative at the frequency location

fm = xi at time i as:

Ii(xi, k) =

Ntx∑

q=1

piq(k)

J∑

j=1

bj(xi)wq,ji,k , (5.7)

dIi(xi, k)

dxi=

Ntx∑

q=1

piq(k)

J∑

j=1

b′j(xi)wq,ji,k . (5.8)

where b′j(f) is the known derivative of the j-th basis function. Expressions (5.7-

5.8) can be used in (3.3) to update the swarming behavior. In wide area ad-hoc

networks, the PU’s communications reach only a subset of SU’s with a high inter-

fering power. In such scenario, estimating the received spectrum through (5.7-5.8)

would enable remote cognitive users to dynamically reuse the PU’s resources.

203

5.3 Adaptive Swarming

5.3 Adaptive Swarming

Combining the adaptive diffusion step (4.3), for estimating and tracking the

interference profile, with the swarming update (3.3), for dynamic resource allo-

cation, we arrive to the following adaptive swarming algorithm to coordinate the

learning and adaptation capabilities of the bio-inspired cognitive network.

Algorithm 1 : Adaptive swarming algorithm

For each node i, start with wi,−1 = 0,ψi,−1 = 0, xi[0] ∈ [fmin, fmax]. Every

node i then performs the following steps for k ≥ 0:

1. The node knows the position xi[k] of its resource on the frequency domain

and has access to the local data di,k,Bi,k.

2. Perform an adaptation step to adaptively estimate the weight vector wo as:

ψi,k = wi,k−1 + µiH i,kBTi,k[di,k −Bi,kwi,k−1], (5.9)

wi,k =∑

j∈Ni

cijψj,k. (5.10)

3. Compute estimates of the interference spectrum and of its derivative in the

current position xk at time i as:

Ii(xi, k) =

Ntx∑

q=1

piq(k)

J∑

j=1

bj(xi)wq,ji,k , (5.11)

dIi(xi, k)

dxi=

Ntx∑

q=1

piq(k)J∑

j=1

b′j(xi)wq,ji,k . (5.12)

4. Update the position of resource i on the frequency axis as:

xi[k + 1] = xi[k]− αi[k]

dIi(xi, k)

dxi−

M∑

j=1

aij g(xj [k]− xi[k])

. (5.13)

204

5.4 Simulation Results


In this section, we provide numerical examples to illustrate the main fea-

tures of the proposed technique combining the swarm-based resource allocation

method, illustrated in Section 4.1, and the distributed cooperative sensing algo-

rithm using ATC diffusion adaptation, shown in Section 4.2.

Numerical Example 1 - Performance : In the following examples, we aim to show

the performance of the cognitive network based on the adaptive swarming algo-

rithm. We consider a connected network composed of 15 SUs, plus the inclusion

of two PUs. The topology of the network is shown in Fig. 5.1, where the SUs

are depicted by dots, whereas the PUs by squares. In particular, a PU moves

from the initial position given by the white square to the final position depicted

by the black square, so that the interference perceived by the secondary network

is time-varying. We consider a polynomial path loss model piq(diq) = (diq/d0)−α

(α = 2), where diq is the distance between the q-th PU and the k-th SU, and

d0 = 1 is a reference distance. The cognitive SU’s scan Nc = 80 channels be-

tween 30 and 45 MHz and use J = 15 Gaussian basis functions to model the

basis expansion of the transmitted spectrum. The single Gaussian basis function

is expressed as:

bj(f) =A

√

2πσ2b

exp

(

−(f − fj)

2

2σ2b

)

. (5.14)

with amplitude normalized to one, and σ2b = 0.5. An example of basis expansion

using Gaussian pulses is shown in Fig. 4.4. To estimate the spectrum profile, we

employ the normalized ATC diffusion algorithm in (4.3) where the step sizes are

set equal to µk = 1, for all k. Furthermore, we consider a combination matrix

C that simply averages the intermediate estimate from the neighborhood, hence,

cj,i = 1/Ni for all l. The white Gaussian noise variance in (4.1) is set equal

to σ2n,i = 0.5 × 10−3, for all i. We assume the presence of 15 resources (to be

allocated) that are initially scattered randomly across the frequency spectrum.

At the k-th iteration of the updating rule (3.3), each node communicates to its

205


0 0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PU

PU

Figure 5.1: Secondary network. The square nodes denote primary users and the

dot nodes denote secondary users.

neighbors the position it intends to occupy, i.e., the scalar xi[k] representing a fre-

quency subchannel. In the application at hand, there is an intrinsic quantization

of the frequency resources given by the subchannel bandwidth. In our implemen-

tation, we let the system evolve according to (3.3) until successive differences in

allocation become smaller than the bandwidth of a frequency subchannel. At that

point, the evolution stops and every SU is allowed to transmit over the selected

channel. We consider an interference profile as in Fig. 5.2, where the dashed curve

depicts the true transmitted spectrum, whereas the solid and dot dashed curves

represent, respectively, the estimation at convergence through ATC diffusion and

without cooperation among nodes. We notice how diffusion adaptation fits well

the spectrum profile while the non-cooperative approach leads to poor estimation.

To evaluate the performance of the distributed estimation technique, in Fig. 5.3

we show the steady-state MSD of the ATC diffusion algorithm compared with the

theoretical result in (4.73). The steady-state values are obtained by averaging

206


30 35 40 45−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Frequency (Mhz)

PS

D (

mW

/Hz)

True Spectrum

ATC Diffusion

No Cooperation

Figure 5.2: Comparison of the result of spectrum estimation through cooperative

diffusion adaptation and without cooperation among the users.

over 200 independent experiments and over 100 time samples after convergence.

It can be observed that the simulation results match well the theoretical values.

An example of resource allocation is shown in Fig. 5.2, where the dots on the

frequency axis represent the final frequency sub-channels chosen at convergence

by the network nodes. The parameters of the swarm are cA = 0.025, cR = 0.25.

We also considered a fixed step size ε0 = 0.05, thus leading to a simple gradient

descent version of the algorithm in (3.3). It is evident how the resources avoid the

positions occupied by primary users, tend to keep the spread as small as possible

while avoiding collisions among the allocations of different users. Observe that

the number of allocated channels is less than the number of requested resources.

This means that a certain number of nodes have picked up the same channels.

We have checked numerically via simulations that, by choosing appropriately the

swarm parameters, the final channel allocation does not lead to collisions among

spatial neighbors. This means that the algorithm is capable of implementing a

decentralized mechanism for spatial reuse of frequencies.

To measure the effectiveness of the distributed resource allocation strategy, in

207


2 4 6 8 10 12 14−40

−38

−36

−34

−32

−30

Node index

MS

D (

dB

)

Simulation results

Theoretical values

Figure 5.3: Steady-state MSD versus node index.

Fig. 5.4 we report the interference level, versus the number of nodes composing

the secondary network, perceived over the frequency slots occupied by the SUs,

after convergence. The result is averaged over 200 independent realizations. We

considered two different values of the receiver noise power σ2n, which determines

the variance of the estimation noise vmi (k) in (4.1). The parameters of the swarm

are cA = 0.025, cR = 0.25 and the interference profile is the same considered in

Fig. 5.2. From Fig. 5.4, we notice that, using a non cooperative approach, the

estimation of the interference profile gradient is quite poor and some resources

end up being allocated by mistake in the regions occupied by the primary users,

trapped because of the estimation errors affecting the algorithm. This explains

the high level of interference perceived in this case. The performance of the allo-

cation can be remarkably improved adopting the cooperative diffusion adaptation

approach. Indeed, as the estimation accuracy improves, each resource tends to

move towards the interference-free regions, thus making the overall swarm ex-

perience a smaller total interference. As the number of nodes N increases, the

allocation performance improves because the swarming algorithm exploits a co-

operative capability given by the cohesion force. This intrinsic robustness deter-

mines that the agents, allocated over the low interference bands, tend to form

cohesive blocks that exert an attraction towards the agents trapped by mistake

over the regions of the spectrum occupied by the primary users. Moreover, in the

208


10 12 14 16 18 20 22 24 26 28 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Number of nodes

Avera

ge inte

rfere

nce (m

W)

No Cooperation σ2 = 0.1mW

ATC Diffusion σ2 = 0.1mW

No Cooperation σ2 = 0.5mW

ATC Diffusion σ2 = 0.5mW

Figure 5.4: Average interference perceived by the swarm at convergence, for the

non cooperative estimation case and for adaptive diffusion.

cooperative case, an increase in the number of nodes also improves the estimation

performance, thus simplifying the resource allocation task. From Fig. 5.4, we

also note, as expected, how a stronger noise leads to worst allocation performance

in both cases. Nevertheless, the performance of the cooperative approach is less

sensitive. This means that the performance of the resource allocation based on

the swarming algorithm can be considerably improved if every node cooperates

with its own neighbors to adaptively estimate the interference profile.

Numerical Example 2 - Learning and adaptation : In this example, we aim to

show the learning and adaptation capability of the cognitive network based on

the adaptive swarming algorithm. Natural swarms are adaptive systems whose

individuals cooperate in order to improve their food search capabilities and to

increase their robustness against predators’ attacks. We show next that the

proposed resource allocation increases, as a by-product, the network robustness

against the intrusion of a primary user (predator). We consider again the network

topology depicted in Fig. 5.1, where the two PU’s start emitting at different

209


30 35 40 45−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Frequency (MHz)

PS

D (

mW

/Hz)

True spectrum

ATC Diffusion

30 35 40 45−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Frequency (MHz)

PS

D (

mW

/Hz)

True spectrum

ATC Diffusion

30 35 40 45

0

1

2

3

Frequency (MHz)

PS

D (

mW

/Hz)

True spectrum

ATC Diffusion

30 35 40 45

0

1

2

3

Frequency (MHz)

PS

D (

mW

/Hz)

True spectrum

ATC Diffusion

i = 125 i = 375

i = 625 i = 875

Figure 5.5: Different resource assignments in dynamic environment.

times, thus causing a dynamic change of the occupied spectrum. Our goal is to

test the dynamic response of the network to this changing environment. The

parameters are the same considered in the previous simulation. In Fig. 5.5 we

show an example of spectrum estimation and swarm-based resource assignment

in the case the PU’s interference is dynamic. As before, the dots on the frequency

axis represent the final channels chosen at convergence by the network nodes. At

the beginning of time, the two PU’s are silent, and the first PU starts to transmit

only at the iteration i = 250. The first PU becomes silent at iteration i = 500

while, at the same time, the second PU starts to transmit. After iteration i = 750,

the two PU’s are both transmitting at the same time. In particular, in Fig. 5.5

it is shown the evolution of the spectrum estimation and resource allocation

210


0 200 400 600 800 1000−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

iteration index

MS

D (

dB

)

Figure 5.6: ATC diffusion learning curve, in terms of MSD.

at 4 different time iterations, i = 125, 375, 625, 875. We notice from Fig. 5.5

how diffusion adaptation fits always well the spectrum profile, thus proving good

tracking performance. To give an example of the tracking capability of the ATC

diffusion filter, in Fig. 5.6, we show the learning curve of the algorithm in terms

of MSD. As we can see, ATC diffusion reacts to the changes in the environment,

learning the spectrum profile through local cooperation. At the same way, from

Fig.5.5, we also notice how the swarm reacts to the PU’s activations, avoiding to

select radio channels occupied by primary transmissions. Resorting again with

the swarm analogy, PU’s take now the role of predators whose positions must be

avoided by the swarm individuals. In this context it is reasonable that the swarm

agents closer to the predator’s positions move faster to avoid the dangerous zones.

To allow this adaptive swarming behavior, in this example, we have considered

a time-varying step size that depends on the perceived interference through a

linear scaling function, e.g. εi[k] = ai + biIi(xi, k), for all i, where ai = 0.05 and

bi = 1. As a consequence, the swarm model in (3.3) accelerates the motion of

the resources perceiving a high interference, improving the reaction time needed

211

5.5 Conclusion

0 200 400 600 800 1000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

iteration index

Avera

ge inte

rfere

nce (

mW

)

Figure 5.7: Average perceived interference versus iteration index.

by the algorithm to perform a resource allocation on idle bands in case of a PU’s

activation. To give an example of the time needed by the algorithm to react to

the PU’s intrusions and adjust the resource allocation consequently, in Fig. 5.7

we show the behavior of the instantaneous interference perceived by the swarm

versus the iteration index. The three peaks correspond to the PU’s activation

times. From Fig. 5.7, we notice how the adaptive swarming algorithm needs

only a small number of iterations to leave the regions occupied by the PU’s.

This positive behavior is a consequence of the adaptation of the algorithm to the

perceived interference, determining that resources allocating on high interference

regions move faster due to the increment of the profile gradient and the cohesion

force.

5.5 Conclusion

In this chapter we have proposed a dynamic resource allocation technique

combining a distributed diffusion algorithm, for implementing cooperative sens-

212

5.5 Conclusion

ing, with a swarming technique, for allocating resources in a parsimonious way

(i.e., avoiding unnecessary spread in the frequency domain), yet avoiding col-

lisions. In the swarm analogy, the dynamic interference distribution over the

frequency domain takes the role of the food spatial distribution. Furthermore,

the occupied zones in the resource domain take the role of dangerous regions that

must be avoided by the swarm individuals as fast as possible, while idle bands

represent regions rich of food that the agents have to occupy reducing their speed.

The swarm mechanism includes an attraction force, useful to minimize the spread

over the resource domain, and a repulsion force, useful to avoid collisions among

swarm members. We employ a diffusion adaptation scheme, which estimates and

learn the interference profile through local cooperation, to guide the movement

of the swarm individuals. A mean-square performance analysis for the diffusion

adaptation filter has been derived and simulation results match well with the

theoretical results. Finally, the procedure has been applied to the dynamic re-

source allocation problem in the frequency domain. Numerical results show the

improvement that results in the resource allocation performance due to the coop-

erative estimation of the spectrum. Furthermore, it is shown how the proposed

technique endows the resulting bio-inspired network with powerful learning and

adaptation capabilities.

213

Chapter 6

Concluding Remarks

This dissertation has considered bio-inspired techniques for dynamic radio

access in cognitive radio systems. Both resource allocation and spectrum sensing

tasks have been considered. For the former, it has been proposed a radio access

mechanism that mimics the behavior of a flock of birds swarming in search for food

in a cohesive fashion without colliding with each other; specifically, the method

has been employed for dynamic radio access in the frequency and time-frequency

domains and detailed stability and convergence analysis has been provided, even

in the presence of random disturbances introduced by realistic radio channels.

For the latter, a distributed method based on diffusion adaptation algorithms

has been developed. First, we have introduced a basis expansion model of the

PU’s PSD, then we have proposed a normalized version of the Adapt then Com-

bine (ATC) diffusion algorithm, which enables the network to learn and track

the time-varying interference profile. Convergence and mean-square performance

analysis of the proposed normalized ATC diffusion filter, applied to the spectrum

estimation problem, has also been derived. Finally, we have extended the swarm

based resource allocation to incorporate a real-time distributed technique for

spectrum estimation based on diffusion adaptation. The proposed procedure has

been applied to the dynamic resource allocation problem in cognitive radio, en-

215

6.1 Conclusions

dowing the resulting bio-inspired network with powerful learning and adaptation

capabilities.

6.1 Conclusions

After giving the motivation of the dissertation in Chapter 1, an overview of the

recent advances in the field of bio-inspired signal processing and networking has

been presented.

Chapter 2 has recalled several important theories on which many results of this

dissertation are based. In particular, basic results have been reported from dy-

namical systems theory, distributed nonlinear optimization, stochastic approxi-

mation theory, and graph theory, which have been be largely used in this disser-

tation.

Chapter 3 has proposed a bio-inspired radio access mechanism for cognitive net-

works mimicking the behavior of a flock of birds swarming in search for food in

a cohesive fashion without colliding with each other. The equivalence between

swarming and radio resource allocation is established by modeling the interference

distribution in the resource domain, e.g. frequency and time, as the spatial distri-

bution of food, while the position of the single bird represents the radio resource

chosen by each radio node. The solution is given as the distributed minimization

of a functional, borrowed from social foraging swarming models, containing the

average interference plus repulsion and attraction terms that help to avoid con-

flicts and maintain cohesiveness, respectively. A stability and cohesion analysis is

derived under different assumptions on the attraction/repulsion terms, showing

the effect played by the swarm parameters and connectivity on the final swarm

size. Several examples have illustrated how the proposed method can be applied

to dynamic resource allocation on the frequency domain and the time-frequency

domain, providing an intrinsic capability of the system to provide spatial reuse

216

6.1 Conclusions

of frequency, through a purely decentralized mechanism. In the last part of the

chapter, we have also considered the swarming algorithm in the presence of chan-

nel imperfections, such as link failures, estimation errors, and quantization noise.

Thus, we have derived the almost sure convergence of the swarming procedure to

an equilibrium configuration dependent on the mean graph of the network, even

in the presence of such random disturbances.

Chapter 4 has developed adaptive methods for spectrum estimation in cognitive

radio networks based on diffusion adaptation algorithms. In particular, it has

been addressed this task through a parsimonious basis expansion model of the

PSD in frequency. This model reduces the sensing task to estimating a common

vector of unknown parameters. The resulting estimator relies on diffusion adapta-

tion algorithms, where the cognitive radios exchange information locally only with

their one-hop neighbors, eliminating the need for a fusion center. First, we have

described the basic diffusion algorithm, then we have introduced novel regular-

ized diffusion LMS techniques for distributed estimation over adaptive networks,

which are able to exploit sparsity in the underlying system model. Convergence

and mean square analysis of the sparse adaptive diffusion filter have shown un-

der what conditions we have dominance of the proposed method with respect to

its unregularized counterpart in terms of steady-state performance. Simulation

results have also confirmed the potential benefits of the proposed filter under the

sparsity assumption on the true coefficient vector. Exploiting these estimation

schemes, we have illustrated the proposed distributed spectrum estimation tech-

nique based on diffusion adaptation. We have first introduced a basis expansion

model, which is useful to model the PU’s transmission, allowing distributed co-

operative sensing. Then, we have proposed a normalized version of the Adapt

then Combine (ATC) diffusion algorithm, which enables the network to learn

and track the time-varying interference profile. Convergence and mean-square

performance analysis of the proposed normalized ATC diffusion filter, applied to

the spectrum estimation problem, has also been derived.

217

6.1 Conclusions

Chapter 5 has studied the learning abilities of adaptive networks in the context of

cognitive radio networks and investigated how well they assist in allocating power

and communications resources in the frequency domain. The allocation mech-

anism is based on a social foraging swarm model that lets every node allocate

its resources (power/bits) in the frequency regions where the interference is at a

minimum while avoiding collisions with other nodes. We have employed adaptive

diffusion techniques to estimate the interference profile in a cooperative manner

and to guide the motion of the swarm individuals in the resource domain. The

resulting bio-inspired network cooperatively estimates the interference profile in

the resource domain of a cognitive network and allocates resources through purely

decentralized mechanisms. Finally, the resulting procedure has been applied to

the dynamic resource allocation problem in the frequency domain. Numerical

results have shown the improvement that results in the resource allocation per-

formance due to the cooperative estimation of the spectrum. Furthermore, it

has been shown how the proposed technique endows the resulting bio-inspired

network with powerful learning and adaptation capabilities.

Summarizing, the main results obtained in this dissertation are:

• the application of swarming mechanisms to radio resource allocation in

cognitive radios;

• the stability and cohesion analysis of social foraging swarms, applied to

the resource allocation problem in cognitive radios, which provides upper

and lower bounds on the spread of the swarm, as a function of the swarm

connectivity;

• fast versions of the swarming algorithm, useful for our application, and the

application of such procedures to the dynamic resource allocation in the

frequency domain;

• the application of the proposed procedure to resource allocation in the time-

frequency domain, where the primary users in a cognitive radio system are

218

6.1 Conclusions

modeled as statistically independent homogeneous continuous-time Markov

processes;

• the derivation of the convergence properties of the proposed algorithms in

the presence of random disturbances such as link failures, quantization noise

and estimation errors;

• the application of compressive sensing techniques in the distributed estima-

tion problem over adaptive networks;

• the derivation of mean-square analysis for the sparse diffusion adaptive

filter;

• a real-time distributed spectrum estimation technique based on diffusion

adaptation;

• the derivation of the mean-square properties of the diffusion adaptive filter

applied to the spectrum estimation problem;

• the extension of the basic social foraging swarming model to incorporate

a real-time distributed spectrum estimation technique based on diffusion

adaptation.

219

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