In this paper, we propose a simple model that can generate small-world network with community structure. The network is introduced as a tunable community organization with parameter r, which is directly measured by the ratio of inter-to intra-community connectivity, and a smaller r corresponds to a stronger community structure. The structure properties, including the degree distribution, clustering, the communication efficiency and modularity are also analysed for the network. In addition, by using the Kuramoto model, we investigated the phase synchronization on this network, and found that increasing the fuzziness of community structure will markedly enhance the network synchronizability; however, in an abnormal region (r ≤ 0.001), the network has even worse synchronizability than the case of isolated communities (r = 0). Furthermore, this network exhibits a remarkable synchronization behaviour in topological scales:the oscillators of high densely interconnected communities synchronize more easily, and more rapidly than the whole network.
