Moment-based analysis of synchronization in small-world networks of oscillators

In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distribution of the Laplacian matrix of the network. We introduce, for the first time, analytical expressions for the first three moments of the eigenvalue distribution of the Laplacian matrix as a function of the probability of shortcuts and the connectivity of the underlying nearest-neighbor coupled ring. We apply these expressions to estimate the spectral support of the Laplacian matrix in order to predict synchronization in small-world networks. We verify the efficiency of our predictions with numerical simulations.

[1]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[2]  G. Bard Ermentrout.  Review of Sync : The Emerging Science of Spontaneous Order , 2003 .

[3]  Guanrong Chen,et al.  A time-varying complex dynamical network model and its controlled synchronization criteria , 2004, IEEE Trans. Autom. Control..

[4]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[5]  A. Mikhailov,et al.  Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems , 2004 .

[6]  Mario di Bernardo,et al.  Effects of Degree Correlation on the Synchronization of Networks of oscillators , 2007, Int. J. Bifurc. Chaos.

[7]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[8]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[9]  Jean-Jacques E. Slotine,et al.  A Study of Synchronization and Group Cooperation Using Partial Contraction Theory , 2004 .

[10]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[11]  Guanrong Chen,et al.  A time-varying complex dynamical network model and its controlled synchronization criteria , 2005, IEEE Transactions on Automatic Control.

[12]  Victor M. Preciado.  Spectral analysis for stochastic models of large-scale complex dynamical networks , 2008 .

[13]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  John Odentrantz,et al.  Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues , 2000, Technometrics.

[16]  Daizhan Cheng,et al.  Characterizing the synchronizability of small-world dynamical networks , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[17]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[18]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[19]  L. Asz.  Random Walks on Graphs: a Survey , 2022 .

[20]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[21]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[22]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[23]  David Malakoff,et al.  U.S. Science Advocate George Brown Dies , 1999, Science.

[24]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[25]  Chai Wah Wu.  Synchronization in arrays of coupled nonlinear systems: passivity circle criterion and observer design , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).