Synchronization in Complex Systems

Synchronization
in
Complex Systems
M. Biey,
Politecnico di Torino, 2016
June 6, 2016
M. Biey - Department. of Electronics and Telecommunications Politecnico di Torino
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Introduction
In 2002, Barahona and Pecora [1] applied the Master
Stability Function formalism to analyzing the
synchronizability of small-world networks.
This marks the beginning of an explosively growing area
of research in synchronization in complex networks (see
[2]-[9]).
[1] M. Barahona, L.M. Pecora, Synchronization in Small-World Systems, Phys.
Rev. Lett., Vol. 89, 054101, 2002.
June 6, 2016
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A measure of synchronizability
A relevant contribution in determining the (local) stability of the synchronized
states was given in by Pecora and Carroll in 1998 [*] by using the eigenvalues
of the Laplacian matrix representing the network.
A brief summary:
Consider a network of N identical dynamical systems with symmetric linear
diffusive coupling. The equations of motion for the system are
N
xi  f (xi )    Gij Hx j , i  1, 2,
,N
(1)
j 1
where f ( ) governs the dynamics of each isolated node, H( ) is a constant
matrix coupling function, ε > 0 is the overall coupling strength, and G is the
Laplacian matrix associated to the network.
[*] L.M. Pecora, T.L. Carroll, Master Stability Functions for Synchronized
Coupled Systems, Phys. Rev. Lett., Vol. 80, pp. 2109-2112, 1998.
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A measure of synchronizability (cont.)
The network admits a synchronous state x1 = x2 =
= xN = s,
whose (local) stability is determined by the corresponding system of variational equations, together
with the motion on the synchronous manifold
s  f (s)
(2)
This system of variational equations can be diagonalized into N blocks of the form
θ  ( Df (s)   DH (s))θ
(3)
where θ represents a mode of perturbation from the synchronized state; Df(s) denotes the Jacobian
matrix of f evaluated at s and DH(s) denotes the Jacobian of matrix of H evaluated at s;
γ = ελi, i = 1, ··· , N; λ1 = 0 >λ2 ≥ ··· ≥ λN are the eigenvalues of G, all of them real as the
matrix G is symmetric.
Λ(γ ) is defined as the largest Lyapunov exponent of the system
defined by Eqs. (2) and (3) as a function of γ. This function, which generally is obtained by numerical
The master stability function (MSF)
methods, determines the stability of the synchronized state. In particular, the synchronized state is
unstable if Λ(ελi) > 0, for at least an index i ∈ {2, . . . , N}.
June 6, 2016
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A measure of synchronizability (cont.)
The region S   where (α )  0
is called synchronization region.
In the majority of cases, S may
have one of the following form:
• S1 = 0 (empty set)
• S2 = (m, + )
• S3 = j (jm, jM)
αm
αM
synchronization
region
In general, j = 1 and m< 0, M < 0
The condition of stable synchronous
state are:
S2 : ελ2  m ;
S3 : a value of ε exists iff λN / λ2 < M / m
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A measure of synchronizability (cont.)
In other words, for a large class of oscillators there exist two
classes of networks:
class-A networks: for which the condition of stable
synchronous state is
ελ2  m = a;
class-B networks: for which this condition reads
λN / λ2 < M / m = b;
a and b depends on f (·) , the synchronous
state s, and the coupling matrix H;
a and b do not depend on the Laplacian matrix G.
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A measure of synchronizability (cont.)
The ratio Q = λN/λ2 depends only on the network topology, while the ratio
αM/αm depends on the dynamics of each node (f ( )) , the synchronous
state and the coupling function (H( )). The lower the Q, the wider the
interval of possible coupling strength ε such that the corresponding
network has a locally stable synchronous state.
As a consequence, building networks which are highly synchronizable
generally means building networks with a low Q.
In practice, the MSF is not always negative only in a finite interval.
However, synchronizability in a variety of dynamical processes can be
described by similar spectral properties, which tend to improve when the
network is more homogeneous. However, simply put, other measures of
synchronizability often go ‘‘hand in hand’’ with λN/λ2.
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Classical Random Networks
Classical random networks (Erdős - Rényi model (*))
Given N vertices, each edge is independently chosen
with probability q :
G(N,q)
Erdős and A. Rényi, “On random graphs,'' Publ. Math Debrecen, vol. 6,
pp. 290-297, 1959.
(*) P.
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Classical Random Networks (cont.)
For Classical Random Networks G(N,q), for large N,
λ2  N, while λN / λ2 approaches 1 [*].
For class-A networks the condition for
synchronization (ελ2 a) reads ε > a/N and ε can be
chosen arbitrary small.
For class-B networks with b > 1, since λN / λ2
approaches 1 for large N, it follows that the
network almost surely synchronizes.
[*] M. Fidler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal,
Vol. 23, pp. 298-305, 1973.
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Classical Random Networks (cont.)
Theorem : Let G(N, q) be a random graph on N
vertices. Then
the class-A network G(N, q) asymptotically
almost surely synchronize for arbitrary small
coupling ε;
the class-B network G(N, q) asymptotically
almost surely synchronize for b > 1.
A proof can be found in [9]
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Conclusions
For a large class of oscillators there exist two
classes of networks with known conditions for
the existence of stable synchronous state,
namely class-A networks and class-B networks
When oscillators belonging to these classes are
used to form networks with classic random
graph on N vertices, then the synchronization
behaviour can be predicted.
More detailed results and proofs can be found
in Ref. [9].
June 6, 2016
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References
[1] M. Barahona, L.M. Pecora, Synchronization in Small-World Systems, Phys. Rev. Lett., Vol. 89,
054101, 2002.
[2] T. Nishikawa, A.E. Motter, Y.C. Lai,1, and F.C. Hoppensteadt, Heterogeneity in Oscillator Networks:
Are Smaller Worlds Easier to Synchronize?, Phys. Rev. Lett., Vol. 91, 014101, 2003.
[3] M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel, and S. Boccaletti, Synchronization is
Enhanced in Weighted Complex Networks, Phys. Rev. Lett., Vol. 94, 218701, 2005.
[4] L. Donetti, P.I. Hurtado, and M.A. Munoz, Entangled Networks, Synchronization, and Optimal Network
Topology, Phys. Rev. Lett., Vol. 95, 188701, 2005.
[5] C. Zhou, A.E. Motter, and J. Kurths, Universality in the Synchronization of Weighted Random
Networks, Phys. Rev. Lett., Vol. 96, 034101, 2006.
[6] C. Zhou, J. Kurths, Dynamical Weights and Enhanced Synchronization in Adaptive Complex
Networks, Phys. Rev. Lett., Vol. 96, 164102, 2006.
[7] L. Huang, K. Park, Y.-C. Lai, L. Yang, and K. Yang, Abnormal Synchronization in Complex Clustered
Networks, Phys. Rev. Lett., Vol. 97, 164101, 2006.
[8] D.-H. Kim, A.E. Motter, Ensemble Averageability in Network Spectra, Phys. Rev. Lett., Vol. 98,
248701, 2007.
[9] M. Biey, F. Corinto, I. Mishkovski, and M. Righero, Synchronization in complex networks: properties
and tools, in L. Kocarev (Editor), Consensus and Synchronization in Complex Networks, Springer-Verlag,
Heidelberg, pp. 111-153, 2013.
June 6, 2016
M. Biey - Dip. di Elettronica Telecomunicazioni - Politecnico di Torino
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