Phase synchronization in complex networks with decayed long-range interactions

We study phase synchronization in random sparse complex networks of small-world and scale-free topologies with long-range decayed couplings, and observe some special features of synchronization different from those in previous studies of all-to-all networks: when the coupling exponent α < α c , the critical exponent; the density of long-range edges improves the synchronous behaviors of networks; and it seriously determines the dependence of network synchronization on the network heterogeneity of scale-free networks.