Synchronization in Generalized Erdös-Rényi Networks of Nonlinear Oscillators

In this paper, we study synchronization of complex random networks of nonlinear oscillators, with specifiable expected degree distribution. We review a sufficient condition for synchronization and a sufficient condition for desynchronization, expressed in terms of the eigenvalue distribution of the Laplacian of the graph and the coupling strength. We then provide a general way to approximate the Laplacian eigenvalue distribution for the case of large random graphs produced by a generalization, [2], of the Erdös-Rényi model. Our approach is based on approximating the moments of the eigenvalue density function. The analysis is illustrated by using a complex network of nonlinear oscillators, with a power-law degree distribution.

[1]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[2]  Bruno O. Shubert,et al.  Random variables and stochastic processes , 1979 .

[3]  Carroll,et al.  Short wavelength bifurcations and size instabilities in coupled oscillator systems. , 1995, Physical review letters.

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

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

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

[7]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[9]  Xiao Fan Wang,et al.  Synchronization in scale-free dynamical networks: robustness and fragility , 2001, cond-mat/0105014.

[10]  Chai Wah Wu,et al.  Synchronization in Coupled Chaotic Circuits and Systems , 2002 .

[11]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[12]  Kevin M. Passino,et al.  Stability analysis of swarms , 2003, IEEE Trans. Autom. Control..

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

[14]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[15]  Luc Moreau,et al.  Stability of multiagent systems with time-dependent communication links , 2005, IEEE Transactions on Automatic Control.

[16]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[17]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.