computer and automation research institute hungarian academy of sciences generation of robust...
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Computer and Automation Research InstituteComputer and Automation Research Institute
Hungarian Academy of SciencesHungarian Academy of Sciences
Generation of Robust Networks Generation of Robust Networks with Optimization under Budget with Optimization under Budget
ConstraintsConstraints
(plus ongoing work)(plus ongoing work)
László GulyásMTA SZTAKI
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
2
AgendaAgenda
• Background– Engineering– Agent-Based Simulation– Modeling Complex Social Systems/Networks
• ‘Engineering’ Robust Networks– Past project– A localized, agent-based approach
• Ongoing work (‘Teaser’)– Discrete Choices on (Endogenous) Networks
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
3
BackgroundBackground
• Software Engineering• Multi-Agent Systems• Agent-Based Modeling and Simulation• Complex Social Systems• Social Networks
– Bottom-Up Approach– Generative Social Models
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
4
‘‘Engineering’ Robust NetworksEngineering’ Robust Networks
• Past project (under publication)– Presented at IWES’04– ‘Networked’ version of previous work (at Lyon TI).
• Generative approach:– Agent-based model.– Maximizing agents.– Limited information access– Limited cognitive abilities.
• A bottom-up, localized version of the Preferential Attachment model.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
5
The Robustness of Internet 1/2The Robustness of Internet 1/2
• Random failures of nodes have little effect on the overall connectivity.
– The networks of Internet have a characteristic (“scale-free”) structure.
– The distribution of the#links per node followsa power law.
• #nodes[#links = x] = x-a
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
6
The Robustness of Internet 1/2The Robustness of Internet 1/2
• Random failures are extremely likely to effect only weakly connected nodes.
– Drawback: susceptibility to planned attacks.
#nod
es
#links
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
7
Generation of Robust NetworksGeneration of Robust Networks
• Purpose:
– Explanation:• Internet evolved to be robust spontaneously
in a distributed manner.• It is an intriguing question to explain how and why.
– Engineering:• It is of practical interest to be able to generate robust
networks without total top-down control.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
8
Top-Down vs. Bottom-Up ApproachTop-Down vs. Bottom-Up Approach
• The prevailing explanation:– Preferential Attachment Model (Albert&Barabási)
(for the generation of scale-free networks):• Incremental addition of nodes.• Each node has a fixed number of links.• Newcomers attach to existing nodes
with probability proportional to the nodes’ connectivity.
• No bottom-up explanation so far.• I propose an agent-based model capable of producing
robust networks.• Scale-free networks as a special case.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
9
The Model: OverviewThe Model: Overview
• Incremental addition of nodes (agents). • A fixed E number of links per agent.
– Initially: E fully connected nodes.
• Agents maximize their connectivity by linking to the nodes with the highest degrees.– Subject to their information access:– They buy information from a Central Authority (CA),
limited by their personal budget constraints b.
• The price of information:– Independent of the agents in question, but may depend on
the size of the network, according to a pricing scheme (PS).
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
10
Details: Information AccessDetails: Information Access
• Agents have no previous information concerning the network.– Therefore they cannot specify the node they are
interested in.
– However, they can list the nodes they already have knowledge about.
– The CA returns random node not contained by the list, together with its degree.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
11
Details: Budget ConstraintsDetails: Budget Constraints
• Homogenous case: – b = B for all agents.
• Heterogeneous case: – b’s are uniformly distributed in [1, B].
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
12
Details: Pricing SchemesDetails: Pricing Schemes
• Size-Independent:
• PS1: PS(i) = C
• Growing Costs:
• PS2: PS(i) = C*B / i
• Decreasing Costs (‘economies of scale’):
• PS3: PS(i) = i / C
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
13
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
14
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Size-Independent PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
15
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– Growing Costs PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
16
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Homogenous Budget Constraints.– ‘Economies of Scale’ PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
17
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Size-Independent PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
18
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– Growing Costs PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
19
Results: Key FindingsResults: Key Findings
• Various combinations of pricing schemes and budget constraints yield robust networks.– Heterogeneous Budget Constraints.– ‘Economies of Scale’ PS.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
20
Results: OverviewResults: Overview
• All three pricing schemes lead to the over-representation of low-degree nodes.– This bias is stronger with the size-independent and
growing costs PS.
• Homogeneous and heterogeneous budget constraints yield qualitatively similar networks.– Except for the decreasing pricing scheme: ‘star topology’.
(Very robust against random failures, but often undesirable.)
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
21
Comparison to Standard NetworksComparison to Standard Networks
• Erdős-Rényi (‘random density‘) Network:
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
22
Comparison to Standard NetworksComparison to Standard Networks
• Watts-Strogatz (‘Small-World’) Network :
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
23
Special Network TopologiesSpecial Network Topologies
• ‘Scale-Free’ (power law) Networks:– The particular ‘growing costs’ PS is a hyperbolic
function of the number of nodes.• Scale-free networks with both homogenous and
heterogeneous budget constraints.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
24
Special Network TopologiesSpecial Network Topologies
• ‘Scale-Free’ (power law) Networks:– The ‘economies of scale’ PS and heterogeneous
budget constraints also yield to a power law distribution of in-edges.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
25
SummarySummary
• A bottom-up approach to generate robust networks was presented.– Also capable of producing special network
topologies, including scale-free networks.
• Driving force: control over information access.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
26
Ongoing Work…Ongoing Work…
• Discrete Choices on Dynamic, Endogenous Networks
• Background & Motivation:– Rush-hour traffic jams in the Netherlands.– Modeling Residential/Transportation Mode
Choices with Social Influences.– Binary/Multinomial/Nested Choices– Generative, agent-based approach.– Empirical extensions.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
27
Discrete Choices on NetworksDiscrete Choices on Networks
• Econometrics approach: discrete choice theory.
• Principles:– Social Influence– Social Dynamics– Coupled Dynamics– Unknown Social Network/Dynamics
Universality Classes.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
28
FrameworkFramework
• Dynamic Social Discrete Choice Model: (A, C, G, R D)
– A={a1, …, aN} – agents
– C={c1, …, cM} – alternatives
– GAA – interaction network
– R=A G {rij} – decision rules (prob. dist.)
– D:G AG – network dynamics
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
29
Framework: ConstraintsFramework: Constraints
• Social Influence: the agents’ utilities of the alternatives is a linear function of the average choice of their neighbors.
• Rules from Probabilistic Logit Model
• An ‘Ising-type’ model, BUT:– From the point of view of the agents.– We are interested in system behavior as a function of the
network, not as a function of the ‘uncertainty’ (temperature) parameter.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
30
Previous WorkPrevious Work
• M=2, G={full network} (“mean-field” case)– Aoki (1995), Brock & Durlauf (2001):
• Two regimes depending on ‘sensitivity’/’certainty’:– The population is equally split (randomized). (1)
– 100% outcome. (2)
M=2, G={Erdős-Rényi networks} or G={Watts-Strogatz
networks} Dugundji & Gulyás(2003)
The latter (2) of the previous two regimes splits: 100% outcome, (2), only if
The network is fully connected, and Has the small-world property.
M=2, G={Erdős-Rényi network} D={Dynamic exogenous rewiring with prob. q} Gulyás & Dugundji (Unpublished)
Do not alter the qualitative outcome. Even for q=1!
M=3, G={full network} (“mean-field” case) Brock & Durlauf (2002)
Two regimes:Equal split.Three 100% outcomes.
M=3, G={Erdős-Rényi networks} or G={Watts-Strogatz
networks} Gulyás & Dugundji (Forthcoming)
Just like the M=2 case: 100% outcomes only if
The network is fully connected, and Has the small-world property.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
31
Focus: Social DynamicsFocus: Social Dynamics
• Social Dynamics, Dynamic Networks.
• Exogenous changes don’t make much difference.– Equal split or 100% dominance.
• In contrast, the real world produces cycles.
• Intuition: Endogenous network dynamics.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
32
Endogenous DynamicsEndogenous Dynamics
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
33
The Endogenous Network Model – The Endogenous Network Model – Binary CaseBinary Case
• u[0,1]: prob. of change per agent, per step.
• zi[0,1]: ratio of same-decision neighbors.
• di[0,N-1]: number of same-dec. neighbors.
di=
0 1 zi
+L
-L
T
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
34
The Endogenous Network Model – The Endogenous Network Model – Binary Case (cont.)Binary Case (cont.)
di defines a class of ‘future networks’.– Probabilistic [uniform] choice.
• Subject to keeping network density constant:– Each new neighbor ‘costs’ one link to the
opposite group.
• Technical constraints:– Non-multiplex network.– Sufficient number of opposite-decision links.
di may only partially be fulfilled.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
35
Preliminary ResultsPreliminary Results
• Initial network: – Erdős-Rényi (random) networks.
• Uniform initial choice distribution:– Only positive feedback in D. (T=1.0)– The effect of the speed of the dynamics (u).– Threshold systems (negative feedback). (T<1.0)
• Biased initial choice distribution:– The “identification problem”.– The role of negative feedback.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
36
Preliminary ResultsPreliminary Results
• Initial network: – Erdős-Rényi (random) networks.
• Uniform initial choice distribution:– Only positive feedback in D. (T=1.0)– The effect of the speed of the dynamics (u).– Threshold systems (negative feedback). (T<1.0)
• Biased initial choice distribution:– The “identification problem”.– The role of negative feedback.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
37
Summary of Threshold SystemsSummary of Threshold Systems
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
38
Preliminary ResultsPreliminary Results
• Initial network: – Erdős-Rényi (random) networks.
• Uniform initial choice distribution:– Only positive feedback in D. (T=1.0)– The effect of the speed of the dynamics (u).– Threshold systems (negative feedback). (T<1.0)
• Biased initial choice distribution:– The “identification problem”.– The role of negative feedback.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
39
Preliminary Experiments with Preliminary Experiments with Biased Initial NetworksBiased Initial Networks
• Positive feedback only (in network dynamics) is not enough to to tip the steady balance.
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
40
Preliminary Experiments with Preliminary Experiments with Biased Initial Networks (cont.)Biased Initial Networks (cont.)
• Negative feedback (T<1) and maybe uneven initial choice distribution seem to be capable of inducing dynamics.
• However, 100% outcomes seem to be extremely hard to achieve.
• Cycles, just like in the real world?
Computer and Automation Research InstituteComputer and Automation Research Institute, , Hungarian Academy of SciencesHungarian Academy of Sciences
EXYSTENCE Thematic Institute
41
Closing WordsClosing Words
• Past and ongoing work on generative, agent-based models of social networks.
• A bottom-up model of network formation.• Understanding the effect of various networks
topologies on the global performance of a ‘well-understood’ model.
• Understanding the effect of dynamic, endogenous networks.