Characterizing the synchronizability of small-world dynamical networks

Many real-world complex networks-display a small-world feature-a high degree of clustering and a small average distance. We show that the maximum synchronizability of a network is completely determined by its associated feedback system, which has a precise meaning in terms of synchronous communication. We introduce a new concept of synchronizability matrix to characterize the maximum synchronizability of a network. Several new concepts, such as sensitive edge and robust edge, are proposed for analyzing the robustness and fragility of synchronization of a network. Using the knowledge of synchronizability, we can purposefully increase the robustness of the network synchronization and prevent it from attacks. Some applications in small-world networks are also discussed briefly.