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Abstract: In this paper, the investigation is first motivated by showing two examplesof simple regular symmetrical graphs, which have the same structuralparameters, such as average distance, degree distribution and node betweennesscentrality, but have very different synchronizabilities. This demonstrates thecomplexity of the network synchronizability problem. For a given network withidentical node dynamics, it is further shown that two key factors influencingthe network synchronizability are the network inner linking matrix and theeigenvalues of the network topological matrix. Several examples are thenprovided to show that adding new edges to a network can either increase ordecrease the network synchronizability. In searching for conditions under whichthe network synchronizability may be increased by adding edges, it is foundthat for networks with disconnected complementary graphs, adding edges neverdecreases their synchronizability. This implies that better understanding andcareful manipulation of the complementary graphs are important and useful forenhancing the network synchronizability. Moreover, it is found that anunbounded synchronized region is always easier to analyze than a boundedsynchronized region. Therefore, to effectively enhance the networksynchronizability, a design method is finally presented for the inner linkingmatrix of rank 1 such that the resultant network has an unbounded synchronizedregion, for the case where the synchronous state is an equilibrium point of thenetwork.

Author: Zhisheng Duan, Guanrong Chen, Lin Huang


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