signal processing in complex networks : many faces of
TRANSCRIPT
Warsaw University of TechnologyFaculty of Physics
Koszykowa 75, PL-00-662 Warsaw, PolandTel: (48) 22 6607267; fax: (48) 22 6282171; http://www.if.pw.edu.pl
Wojciech Zduniak Wojciech Zduniak andandAndrzej KrawieckiAndrzej Krawiecki
SignalSignal processingprocessing inin complexcomplex networksnetworks: : many many facesfaces ofof
structuralstructural stochasticstochastic multiresonancemultiresonance
SummerSummer SolsticeSolstice 2011 2011 InternationalInternational ConferenceConference on on DiscreteDiscrete ModelsModels ofof ComplexComplex SystemsSystems
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Stochastic resonance consists in the maximization of the response of a (nonlinear) system to a periodic stimulation by addition of noise at the input [R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A14, L453 (1981).]
( )ttDtt
xxxV
ttAdx
dV
dt
dxs
′−=′
+−=
++−=
δξξ
ξω
2)()(41
21
)(
)(sin
42
Example Overdamped particle in a bistable potential well
For optimum noiseintensity D the position ofthe particle in the twopotential wells issynchronized with theperiodic signal
Escape rate from theright or left potential wellis given by the Kramersformula and isasymetrically modulatedby the periodic signal
( )
,sin
exp2
1
∆−
=±
D
tAV
tr
s
K
ωπ
m
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
SP(ωωωωS)
SN(ωωωωS)
SPA
>+<−
=0)(dla1
0)(dla1)(
tx
txty
Characterization of stochasticresonance
Output signal
Measures:• Signal-to-noise ratio (SNR)• Spectral power amplification (SPA)
( )( )
( )2
SPA
log10SNR
A
S
S
S
SP
SN
SP
ωωω
=
=
Typical power spectrum density of the outputsignal from a system exhibiting stochasticresonance (top) and dependence of the SNR andSPA on the noise intensity D (bottom).
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• Coupling bistable elements driven by a periodic signal an d noise can lead to the enhancement of stochastic resonance (e.g., array enha nced stochasticresonance in spatially extended systems).
• Can suitably chosen structure of the coupling lead to qualitatively newphenomena in stochastic resonance?
• Example: structural stochastic multiresonance can occur in the Ising model on certain scale-free networks (the curves SPA vs. T show double maxima) [A. Krawiecki, European Phys. J. B69, 81 (2009)].
• Example: structural stochastic multiresonance can also occur in systems ofcoupled passive threshold elements on hierarchical netwo rks , e.g., tree-likenetworks or Ravasz-Barabási networks [M. Kaim and A. Krawiecki, Phys. Lett. A374, 4814 (2010) ].
Stochastic multiresonance (concept): J.M.G. Vilar, J.M. Rubi, Phys. Rev. Lett. 78, 2882 (1997); Physica A264, 1 (1999).
Motivation for the study of stochastic resonance in system s with the structureof complex networks
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• Periodic signal: oscillating magnetic field,• Noise: thermal fluctuations (proportional to the temperature),• Output signal: the time-dependent order parameter (e.g., magnetization).
Stochastic resonance in the Ising model
The Ising model is treated as a complex system which consists of coupled bistableelements (spins), and its response to the periodic signal is studied as a function ofthe temperature and frequency of the magnetic field.
Exemplary results
• SR in the 1-dimensional Ising model (the paramagnetic phase) J.J. Brey and A. Prados, Phys. Lett. A216, 240 (1996); U. Siewert and L. Schimansky-Geier, Phys. Rev. E58, 2843 (1998);
• SR in the 2- and 3-dimensional Ising model (Monte-Carlo simulations and theory in the mean-field approximation): Z. Neda, Phys. Rev. E51, 5315 (1995), K.-T. Leung and Z. Neda, Phys. Lett. A246, 505 (1998) et al.;
• Connection with dynamical phase transitions: B.J. Kim et al., Europhys. Lett. 56, 333 (2001);
• SR in the Ising model on complex networks: H. Hong et al., Phys. Rev. E66, 011107 (2002)(Watts-Strogatz small-world networks), A. Krawiecki, Int. J. Modern Phys. B18, 1759 (2004)(Barabasi-Albert scale-free networks).
• Structural stochastic multiresonance in the Ising model on scale-free networks: A. Krawiecki, Eur. Phys. J. B69, 81 (2009).
Regular lattice Complex network
node
edge
Quantity of interest: distribution of connectivity pk (= probability distribution that a randomly selected node has connectivity k)
Connectivity of the node i: the number of edges attached to the node i
( ) ( )4δδ =∝ zpk
( )
= −
K
γσ
Ak
kG
pk
,
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Complex networks
• Networks with complex topology are ubiquitous in real world . Animportant class of complex networks are scale-free networks whichlook similar at any scale; e.g., the distribution of connectivity k obeys a power scaling law , pk ∝ k-γ.
• Examples of scale-free networks comprise, e.g.,
• the internet activity,
• the www links,
• networks of cooperation (between scientists, actors, etc.),
• traffic networks (airplane & railway connections, city transport schemes),
• biological networks (sexual contacts, protein interactions, certain neuralnetworks), etc.
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Snapshot view ofinternet connections
Imported from:http://www.nd.edu/~networks/gallery.htm
hubs
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Examples of scale-free networks: internet connections
Imported from:http://www.nd.edu/~networks/gallery.htm
Examples of scale-free networks: Map of protein-protein interactions.The colour of a node signifies the phenotypic effect of removing the corresponding protein (red, lethal; green, non-lethal; orange, slow growth; yellow, unknown).
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Hamiltonian
∑ ∑−−=ji i
ijiij sthssJk
H,
00 sin1 ω Jij=J if there is an edge between nodes i,j,
Jij=0 otherwise
The local field acting on spin i
( ) ∑ +=j
jiji thsJk
tI 00 sin1 ω
( ) ( )
−=T
tIssw i
iii tanh12
1
Glauber dynamics (heat bath algorithm)
The order parameter
( ) ∑=i
ii skkN
tS1
( ) ( )∑−
=→∞−==
1
0012
0
2
1 explimT
tT
titSPh
PSPA ω
Spectral Power Amplification (SPA)
The transition rate between two spin configurations which differ by a single flip of one spin, e.g., that in node i
The spins si are located in the nodes i and edges between the nodes correspond to non-zero exchange interactions between the corresponding spins.
Ferromagnetic phase transitionCritical temperature: maximumfluctuations of the order parameter
222 SSS −=δ
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
The Ising model on complex networks
• The algorithm starts with a small number m of fully connected nodes.
• Evolving network: New nodes are added step by step,
• Preferential attachment: From each new node m new links are created to theexisting nodes, and the probability to create an edge to the node i is
where ki is the actual number of edges attached to the node i.
• The network grows until a given number of nodes N is added.
• Evolving network + preferential attachment = scale-free network. The distribution ofthe degrees of nodes is
( ) ( )∑ −>++=i
iii mBBkBkp ;
( ) mBkBkpkk += →+∝ −→∞
− 3, γγγ
[A.-L. Barabási and R. Albert, Science 286, 509 (1999) (for B=0, γ=3); S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)]
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Creating scale-free networks
Signal: S(t)N = 10 000,m = 5,h0= 0.01,(a) B = - 4 (γ =2.2),(b) B = 10 (γ =5)
(a)
(b)
Multiresonance(two maxima of the SPA)
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Structural stochastic multiresonance in the Ising model o n scale-free networks
• It can be seen that the occurrenceof stochastic multiresonancedepends on the structure ofconnections, thus in (a) structuralstochastic multiresonance occurs,• Structural stochasticmultiresonance occurs for 2 < γ < 3provided that the power-law tails ofthe distribution of the degrees ofnodes are fully developed, i.e., no artificial constraint is imposed on the maximum degree.
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
cutting rewiring
Creating coupled scale-free networks
Network ANA nodes,exponent γA
Network BNB nodes,exponent γB
Subnetwork A,Rewiring probability π(A)
NA nodes, exponent γA
Subnetwork B,Rewiring probability π(B)
NB nodes, exponent γB
Rewiring probability (in a given network) = = number of rewired edges (in a given network) / number of edges (in a given network).
Cutting and rewiring do not change the distribution of degrees of nodes in the subnetworks.
For small rewiring probabilities the composite network consists of separate, densely connectedsubnetworks with few inter-subnetwork links, so it has modular structure; For increasing rewiring probabilities the network structure becomes more and more uniform
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
The Ising model on coupled scale-free networksThe spins si are located in the nodes i and edges between the nodes correspond to non-zero exchange interactions between the corresponding spins. The sysyem is subject to the oscillatingmagnetic field
( ) ∑+=
=
=BA NNN
iii sk
kNtS
1
1
Order parametersFor the composite network For the subnetworks
( ) ( ) ∑∑==
==BA N
iii
BBB
N
iii
AAA sk
kNtSsk
kNtS
11
1,
1
• Phase transitions in the Ising model on coupled scale-free networks (Barabási-Albert ones) inthe absence of the magnetic field were considered by K. Suchecki, J.A. Hołyst, Phys. Rev. E74, 011122 (2006), Phys. Rev. E80, 031110 (2009).• In general, there is ferromagnetic transition as the temperature decreases (spins in bothsubnetworks are aligned in parallel), although special preparation of the initial condition can leadto antiparallel alignement of spins in the two subnetworks.• In the analysis based on the mean-field and linear response theory approximations it will be assumed that the ordered stationary state is the ferromagnetic one.
( )2
,
,
2
,
BA
BABAc
k
kJT =
( ) ( )( )( )
( ) ( )( )( )BABA
BABAcBA
NBA
BABA
BABAcBA
JT
NJTBA
BA
BA
,,
2,,
,
1
3
,,,
2,,
,
31
23
31
232
,
,
,
γγγ
γ
γγγ
γ γγ
−−−
=⇒>
∞→−−
−=⇒<<
∞→−
−
Critical temperatures for the uncoupled subnetworks (ferromagnetic transition)
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• The Master equation for theprobability that at timet thesystem is in the spin configuration (s1, s2, ..., sn)
( ) ( ) ( )
( ) ( )tssssPsw
tssssPswtsssPdt
d
nj
N
jjj
nj
N
jjjn
;,,,,,
;,,,,,;,,,
211
211
21
KK
KKK
−−+
+−=
∑
∑
=
=
( ) ( ),tanh1
2
1
−=T
tIssw i
iii( ) ∑ +=
jjiji thsJ
ktI 00 sin
1 ω
• Multiply both sides by si and perform an ensemble average (denoted by ⟨ ⟩), i.e., summation over allpossible spin configurations of the composite network
( )BA
ii
i NNNiT
tIs
dt
sd+==
+−= ,,2,1,tanh K
• The aboveN equations can be formally separated into two groups ofNA andNB equations, for the meanvalues of spins belonging to the sunbentworksA, B, respectively
( )( )
( )( ) ( )( )
( )( )B
BiB
i
Bi
A
AiA
i
Ai
NiT
tIs
dt
sdNi
T
tIs
dt
sdKK ,2,1,tanh;,2,1,tanh =
+−==
+−=
• Make the following mean-field approximation
Mean field approximation
( )( ) ( )( )( ) ( )
thssk
JtItI
Bi
Ai j
jj
jBA
iBA
i 00KK
,, sin ω+
+=≈ ∑∑
∈∈
( )( ) ( )( )
≈
⇒
T
tI
T
tIBA
iBA
i
,,
tanhtanh
Equations of motion for the order parameter
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• Divide the nodes of the subnetworksaccording to their degreesk and assume that the average values ofspins located in the nodes belonging to the class with degree k are equal to ⟨sk⟩• Replace the sums over the nodes of the subnetworkswith sums over the classes of nodes, e.g.
( )( )( )( )
( )( ) ( )( )( )( )
( )( )
( )( ) ( )( )( )( )
( )( ) ( )( )( )
( )( )
( )( ) ( )( ) ( ) ( )( )[ ] ( )( )
( )( ) ( )( ) ( )( ) ( ) ( )( ) ,1~
,
,sin~
sin1
sin1
,
0000
00
maxmax
maxmax
tStStSk
kJJ
thtSk
kJthtStS
k
kJ
thtsk
kpts
k
kp
k
JktI
tsk
kptSts
k
kptS
BAAAAAA
A
A
iABAAA
A
iA
Bk
k
mk B
BkAA
k
k
mk A
AkAiA
i
Bk
k
mk B
BkBA
k
k
mk A
AkA
BA
BA
ππ
ωωππ
ωππ
+−≡≡
+=++−=
=+
+−=
==
∑∑
∑∑
==
==
and similarily for the networkB.In the above equations it was taken into account that the probability that a nodei with degreeki belongingto the subnetworkA is connected to a node with degreek, belonging to
• subnetworkA is • subnetworkB is
and similarily for a nodei belonging to the subnetworkB.
( )( )( )
A
AkA
k
kpπ−1( )
( )
B
BkA
k
kpπ
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• Multiply both sides of the equations of motion for ⟨si(A,B)⟩ by ki , perform the sum over all nodes of the
respective subnetwork and replace it with a sum over the classes of nodes,
( )( )
( )( )( )
( )
( )( )
( )( )( )
( )
∑
∑
=
=
++−=
++−=
B
A
k
mk
B
B
B
B
BkB
B
k
mk
A
A
A
A
AkA
A
tT
htS
Tk
kJ
k
kpS
dt
Sd
tT
htS
Tk
kJ
k
kpS
dt
Sd
max
max
00
00
sin~
tanh
,sin~
tanh
ω
ω
Stationary values of the order parameters and magnetizations for the subnetworks
( )( )
( )( )( )
( )( )
( )( )( )
∑
∑
=
=
=
=
⇒=B
A
k
mk
B
B
B
B
BkB
k
mk
A
A
A
A
AkA
tSTk
kJ
k
kpS
tSTk
kJ
k
kpS
hmax
max
00
00
0~
tanh
,~
tanh
0
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
The case of two coupled scale-free networks
( ) ( )( )( )
( ) ( )( )( ) 1
max1
1
1
max
1
max1
1
1
max
1,,2,
1,,2,
+−+−
−−
+−+−
−−
−−==<=
−−==<=
BB
BB
AA
AA
B
BB
BA
Bk
A
AA
AA
Ak
kmBmNkBkp
kmAmNkAkp
γγγγ
γγγγ
γγ
γγ
Replacing summation with integration, one gets
( )( )( )
( )( )( )
( ) ( )( ) ( )
( )( )( )
( )( )( )
dkTk
SkJBkM
dkTk
SkJAkM
Tk
SkJpM
dkTk
SkJ
k
BkS
dkTk
SkJ
k
AkS
B
B
A
A
A
BB
AA
k
m B
BBB
k
m A
AAA
A
AA
k
mk
Ak
A
k
m B
BB
B
B
k
m A
AA
A
A
∫
∫
∑
∫
∫
=
=→
→
=
=
=
−
−
=
+−
+−
max
max
max
max
max
0
0
0
0
0
0
01
0
01
0
~tanh
~tanh
~tanh
~tanh
,~
tanh
γ
γ
γ
γ
For non-zero rewiring probability there is a commoncritical temperature for both subnetworks Tc. For T<Tc the mean-field stationary values of the order parameters deviate from zero and there is a ferromagnetic transition, but the critical behaviour inthe vicinity of the critical point for both subnetworks canbe different. Below the critical point the magnetizations in bothsubnetworks are parallel.
( ) ( )00
, BA SS
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Linear response theory( )
( )( )
( )( )( )
( )( )
( )( )( )
( )
∑
∑
=
=
++−=
++−=
B
A
k
mk
B
B
B
B
BkB
B
k
mk
A
A
A
A
AkA
A
tT
htS
Tk
kJ
k
kpS
dt
Sd
tT
htS
Tk
kJ
k
kpS
dt
Sd
max
max
00
00
sin~
tanh
,sin~
tanh
ω
ωIn order to solve theequations of motion
( )( ) ( ) ( )( )( ) ( ) ( )
( )( ) ( )( )( ) ( )
( ) ( )
( )( ) ( )( )( ) ( )
( ) ( )
( )( )( )
( )( )( )
∑∑
∑
∑
=
−
=
−
−−
−
=
−
−−
−
=
−
=
=
−
−=
−−=
−
−=
−−=
+−−=
+−−=⇒
+=
+=
BA
B
A
k
mk B
BBB
k
B
B
k
mk A
AAA
k
A
A
BBB
B
BA
k
mk B
BBB
k
B
BBBB
AAA
A
AB
k
mk A
AAA
k
A
AAAA
B
BA
A
BB
BB
A
AB
A
AA
AA
BBB
AAA
Tk
SkJkp
kQ
Tk
SkJkp
kQ
Tk
SkJkp
kT
J
Tk
SkJkp
kT
J
thT
Q
dt
d
thT
Q
dt
d
tStS
tStS
maxmax
max
max
0202
1
1
1
0222
1
1
1
0222
00
00
0
0
~cosh
1,
~cosh
1
11
,~
cosh11
,11
,~
cosh11
sin
sin
τπ
πτπτ
τπ
πτπτ
ωτξ
τξξ
ωτξ
τξξ
ξξ
let us assume that the external magnetic field forces the order parameters of the subnetworks to performsmall oscillations around their stationary values
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
The case of two coupled scale-free networks
( )( )( )
( )( )( )
( ) ( )
( )
( )( )( )( )
( )( )( )
( ) ( ) ( )( )
( )
( )
−+
−
=
−−−+
+
−
−=
+−+−
−
+−+−
AA
A
A
A
AA
A
AAAA
AA
A
AA
A
A
A
AAAA
A
AA
A
A
A
AA
SJ
M
Tk
SmJm
Tk
SkJk
SJ
ATQ
S
S
Tk
SkJk
Tk
SmJm
Sk
A
AA
AA
0
0010max1
max
0
1
0
0
0max2
max02
0
~1~
tanh~
tanh
21~1
~tanh
~tanh
1
γ
γπ
πτ
γγ
γγ
In the ordered state, for T < Tc
In the disordered state, for T > Tc
( )( )( )
( )
1
11 2
21
=
=
−−=
−
A
A
AAc
AcAA
AA
Q
k
kJT
T
T
k
kπτ
The parametersτBB, QB are obtained in a similar way.
(critical temperatue for the subnetworkA in the case with no rewiring)
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Solution of the equations of motion in the linear response approximation is
( ) ( )
( ) ( ) ( )TSPAN
NTSPA
N
NTSPA
hTSPASPA
hTSPASPA
hT
Q
hT
Q
tt
tt
BB
AA
BBAA
B
A
BBBA
BBBA
ABAA
ABAA
B
A
+=
+==+==
=
−
−
+=+=
−−
−−
−−
−−
20
22
21
20
22
21
0
0
2
1
2
1
10
10
11
110
10
1
0201
0201
4,
4
0
0
0
0
0
0
cossin
cossin
ββαα
ββαα
τωτωττ
ττωτωτ
ωβωβξωαωαξ
where the coefficients can be obtained from the system of linear equations
Spectral power amplification SPA as a function of temperature
Spectral power amplification for the subnetworks is
and for the composite network
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Structural stochastic multiresonancein the Ising model on two coupledscale-free networks
Networks with the same number of nodes, NA = NB, and different exponents γA ≠γB.
• Coupling networks with different criticaltemperatures results in the curves SPA vs. Twith two distinct maxima.• As the rewiring probability increases themaxima approcah each other and themaximum at lower temperature graduallydisappears.
Signal: S(t)NA = NB = 5 000,m = 5,ωs=2π/512, h0= 0.01,for different probabilities of rewiringπ (A) = π (B) = ν (seelegends);Subnetwork A:B = 10 (γ A=5),Subnetwork B:B = -3 (γΒ =2.5)(a) Theoretical results in the mean-field and linear responsetheory approximations,(b) Numerical results from Monte Carlo simulations,
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
The same system as on the previous slide, but apart from SPA vs. T for the whole networkalso SPAA,B vs. T for the two subnetworks areshown. • For small rewiring probability the network hasmodular structure and the curves SPAA,B vs. Tresemble those for uncoupled networks. Twomaxima of SPAB vs. T are observed for thenetwork with γB <3.• As the rewiring probability increases and thedensity of inter-subnetworks links becomeshigher the curves SPA vs. T for the twosubnetworks approach each other and double maxima of the SPAA appear also for thenetwork with γA>3,• The curve SPA vs. T for the compositenetwork resembles in general that for thesubnetwork with higher critical temperature.
Black symbols/line –SPA vs. T for the whole network,Blue symbols/line –SPAA vs. T,Red symbols/line –SPAB vs. T,Upper panels: Theoretical results in the mean-field and linearresponse theory approximations,Lower panels: Numerical results from Monte Carlosimulations
π(A) = π(B) =0
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
π(A) = π(B) =0.06 π(A) = π(B) =0.3
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
Networks with different number of nodes, NA ≠ NB, and the same exponents γA =γB.
• Coupling networks with different criticaltemperatures can result in the curves SPA vs. T with two or even three distinct maxima for small to moderate rewiring probability,• Two maxima are located close to the criticaltemperatures for the two uncoupledsubnetworks, and one deep in theferromagnetic phase,• As the rewiring probability rises the twoformer maxima usually merge into one, located close to the higher of the two criticaltemperatures
NA = 3000, NB = 5 000, m = 5, ωs=2π/512, h0= 0.01,π (A) = 0.02, π (B) = 0.012
B = -3 (γA= γΒ =2.5)(a) Theoretical results in the mean-field and linear responsetheory approximations,(b) Numerical results from Monte Carlo simulations,
Black symbols/line –SPA vs. T for the whole network,Blue symbols/line –SPAA vs. T,Red symbols/line –SPAB vs. T.
W. Zduniak and A. Krawiecki Signal processing in complex networks: many faces of structural stochastic multiresonance
Summer Solstice 2011 International Conference on Discrete Models of Complex Systems 6-10 June 2011, Turku, Finland
• Stochastic multiresonance can be observed in the Ising model on scale-free networks with 2 <γ < 3, for small and moderate frequencies of the oscillating magnetic field and for a large enoughnumber of interacting spins N. One maximum of the curve SPA vs. T (a trivial one) appears atT ≈ Tc and the other one usually at T << Tc. .
• The necessary condition for the occurrence of stochastic multiresonance is the presence of thefully developed power-law tails in the distribution of the degrees of nodes pk (structuralstochastic multiresonance).
• Coupling scale-free networks with different critical temperatures (different number of nodes orexponents γ) also leads to the occurrence of structural stochastic multiresonance. Curves SPAvs. T exhibit two or three maxima.
• The maxima of the SPA vs. T for the composite network can be related to those for theuncoupled subnetworks if the rewiring probability is small to moderate and the compositenetwork has modular structure.
• As the rewiring probability increases and the structure of the composite network becomesmore uniform the curve SPA vs. T for the composite network becomes similar to that for thesubnetwork with higher critical tempetrature.
Conclusions