Synchronization In Complex Networks Of Nonlinear Dynamical Systems

By: Chai Wah Wu

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Hardcover | 9 October 2007

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168 Pages

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22.86 x 15.24 x 1.27

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This book brings together two emerging research areas: synchronization in coupled nonlinear systems and complex networks, and study conditions under which a complex network of dynamical systems synchronizes. While there are many texts that study synchronization in chaotic systems or properties of complex networks, there are few texts that consider the intersection of these two very active and interdisciplinary research areas.The main theme of this book is that synchronization conditions can be related to graph theoretical properties of the underlying coupling topology. The book introduces ideas from systems theory, linear algebra and graph theory and the synergy between them that are necessary to derive synchronization conditions. Many of the results, which have been obtained fairly recently and have until now not appeared in textbook form, are presented with complete proofs. This text is suitable for graduate-level study or for researchers who would like to be better acquainted with the latest research in this area.

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Preface	p. vii
Introduction	p. 1
Graphs, Networks, Laplacian Matrices and Algebraic Connectivity	p. 5
Graphs and digraphs	p. 5
Matrices and graphs	p. 8
Algebraic connectivity	p. 18
Undirected graphs	p. 18
Directed graphs	p. 19
Basic properties of a[subscript 1] and b[subscript 1]	p. 21
Basic properties of a[subscript 2]	p. 24
Locally connected graphs	p. 25
Basic properties of a[subscript 3] and a[subscript 4]	p. 27
Examples	p. 27
Hypergraphs	p. 28
Further reading	p. 29
Graph Models	p. 31
Examples of complex networks	p. 31
Neural networks	p. 31
Transportation networks	p. 32
World Wide Web	p. 33
Erdos number and scientific collaboration network	p. 33
Film actor network	p. 33
Citation network	p. 34
Classical random graph models	p. 35
Algebraic connectivity of random graphs	p. 35
Small-world networks	p. 37
Watts-Strogatz model	p. 38
Newman-Watts model	p. 38
Algebraic connectivity of small-world networks	p. 38
Scale-free networks	p. 39
Random geometric graphs	p. 40
Graphs with a prescribed degree sequence	p. 41
Algebraic connectivity and degree sequence	p. 45
Regular graphs	p. 45
Construction 1: graph with low [lambda subscript 2] and r	p. 45
Construction 2: graph with high [lambda subscript 2] and r	p. 47
Graphs with prescribed degree sequence	p. 47
Construction 1: graph with low [lambda subscript 2] and r	p. 48
Construction 2: graph with high [lambda subscript 2] and r	p. 49
Further reading	p. 49
Synchronization in Networks of Nonlinear Continuous-time Dynamical Systems	p. 51
Static coupling topology	p. 53
Properties of [mu](G)	p. 54
Computing [mu](G)	p. 59
Zero row sums matrices	p. 60
Matrices in W	p. 61
Synchronization and algebraic connectivity	p. 62
Coupling topology with a spanning directed tree	p. 62
Time-varying coupling topology	p. 66
Coupling between delayed state variables	p. 67
Choosing the factorization B[subscript 1]B[subscript 2] = UG[subscript tau multiply sign in circle] VD[subscript tau]	p. 69
Choosing the matrix U [epsilon] W[subscript s]	p. 70
Choosing the matrix K	p. 70
The case D = 0	p. 71
Synchronization criteria based on algebraic connectivity	p. 72
Further reading	p. 74
Synchronization in Networks of Coupled Discrete-time Systems	p. 75
Synchronization of coupled scalar maps via contractivity of operators	p. 81
Synchronization in Network of Systems with Linear Dynamics	p. 83
Autonomous coupling	p. 83
Nonautonomous coupling: continuous-time case	p. 84
Slowly varying coupling	p. 85
Nonautonomous coupling: discrete-time case	p. 86
A discrete-time consensus problem	p. 86
Ergodicity of inhomogeneous Markov chains	p. 87
Contractive matrices	p. 96
Pseudocontractivity and scrambling stochastic matrices	p. 98
Set-nonexpansive and set-contractive operators	p. 99
Set-contractivity under the max-norm	p. 103
Set-contractivity under the Euclidean norm	p. 104
Set-contractivity under a weighted Euclidean norm	p. 104
Set-contractivity and coefficient of ergodicity	p. 108
Further reading	p. 109
Agreement and Consensus Problems in Groups of Interacting Agents	p. 111
Continuous-time models	p. 111
Rate of exponential convergence	p. 113
Dynamic coupling topology	p. 113
Discrete-time models	p. 114
Follow the leader dynamics and leadership in coordinated agents	p. 118
A nonlinear model for consensus	p. 119
Agreement in random networks	p. 120
Synchronization in random networks without the scrambling condition	p. 122
Further reading	p. 123
Algebraic Connectivity and Combinatorial Properties of a Graph	p. 125
Properties of algebraic connectivity	p. 125
Vertex and edge connectivity	p. 135
Graph partitions	p. 135
Maximum directed cut	p. 135
Edge-forwarding index	p. 136
Bisection width	p. 136
Isoperimetric number	p. 137
Minimum ratio cut	p. 138
Independence number	p. 138
Semibalanced graphs	p. 139
Bibliography	p. 143
Index	p. 153
Table of Contents provided by Ingram. All Rights Reserved.

Synchronization In Complex Networks Of Nonlinear Dynamical Systems

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