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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.