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Network synchronizability analysis: a graph-theoretic approach. (English) Zbl 1309.05166

Summary: This paper addresses the fundamental problem of complex network synchronizability from a graph-theoretic approach. First, the existing results are briefly reviewed. Then, the relationships between the network synchronizability and network structural parameters (e.g., average distance, degree distribution, and node betweenness centrality) are discussed. The effects of the complementary graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network synchronizability is established. Several examples are given to show that adding new edges to a network can either increase or decrease the network synchronizability. To that end, some new results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same synchronizability is derived. Finally, some examples on Chua circuit networks are presented for illustration.{
©2008 American Institute of Physics}

MSC:

05C82 Small world graphs, complex networks (graph-theoretic aspects)
34D06 Synchronization of solutions to ordinary differential equations
34B45 Boundary value problems on graphs and networks for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37N35 Dynamical systems in control
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