Synchronization in random networks with given expected degree sequences

Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N !1 . In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchronize.

[1]  C. Wu.  Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems , 2003, nlin/0307052.

[2]  T. Carroll,et al.  MASTER STABILITY FUNCTIONS FOR SYNCHRONIZED COUPLED SYSTEMS , 1999 .

[3]  Béla Bollobás,et al.  Random Graphs , 1985 .

[4]  Linyuan Lu,et al.  Random evolution in massive graphs , 2001 .

[5]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[6]  Guanrong Chen,et al.  Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint , 2003 .

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

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

[9]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[11]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[12]  Rabinder N Madan,et al.  Chua's Circuit: A Paradigm for Chaos , 1993, Chua's Circuit.

[13]  Alan M. Frieze,et al.  A general model of web graphs , 2003, Random Struct. Algorithms.

[14]  Johnson,et al.  Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Albert-László Barabási,et al.  Linked: The New Science of Networks , 2002 .

[16]  Wilson J. Rugh,et al.  Linear system theory (2nd ed.) , 1996 .

[17]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[18]  Changsong Zhou,et al.  Universality in the synchronization of weighted random networks. , 2006, Physical review letters.

[19]  Béla Bollobás,et al.  The Isoperimetric Number of Random Regular Graphs , 1988, Eur. J. Comb..

[20]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[21]  Joel E. Cohen,et al.  Community Food Webs: Data and Theory , 1990 .

[22]  M. Hasler,et al.  Connection Graph Stability Method for Synchronized Coupled Chaotic Systems , 2004 .

[23]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[24]  M. Hasler,et al.  Blinking model and synchronization in small-world networks with a time-varying coupling , 2004 .

[25]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[26]  Eli Upfal,et al.  Stochastic models for the Web graph , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[27]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[28]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[29]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[30]  Martin H. Levinson.  Linked: The New Science of Networks , 2004 .

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

[32]  Norman Biggs.  Algebraic Graph Theory: Index , 1974 .

[33]  W. Rugh.  Linear System Theory , 1992 .

[34]  Andrei Z. Broder,et al.  Graph structure in the Web , 2000, Comput. Networks.

[35]  Bojan Mohar,et al.  Laplace eigenvalues and bandwidth-type invariants of graphs , 1993, J. Graph Theory.

[36]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

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

[38]  Neo D. Martinez,et al.  Simple rules yield complex food webs , 2000, Nature.

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

[40]  Ljupco Kocarev,et al.  Sporadic driving of dynamical systems , 1997 .

[41]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[42]  Paul Erdös,et al.  On random graphs, I , 1959 .

[43]  Linyuan Lu,et al.  The Small World Phenomenon in Hybrid Power Law Graphs , 2004 .

[44]  Chris Arney.  Sync: The Emerging Science of Spontaneous Order , 2007 .

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

[46]  C. Wu.  Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling , 2003 .

[47]  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).

[48]  Charles M. Newman,et al.  Community Food Webs , 1990 .

[49]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[50]  Ljupco Kocarev,et al.  On the synchronization region in networks of coupled oscillators , 2004, 2004 IEEE International Symposium on Circuits and Systems (IEEE Cat. No.04CH37512).

[51]  Michalis Faloutsos,et al.  On power-law relationships of the Internet topology , 1999, SIGCOMM '99.

[52]  Guanrong Chen.  Controlling Chaos and Bifurcations in Engineering Systems , 1999 .

[53]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .