Modular synchronization in complex networks with a gauge Kuramoto model

We modify the Kuramoto model for synchronization on complex networks by introducing a gauge term that depends on the edge betweenness centrality (BC). The gauge term introduces additional phase difference between two vertices from 0 to π as the BC on the edge between them increases from the minimum to the maximum in the network. When the network has a modular structure, the model generates the phase synchronization within each module, however, not over the entire system. Based on this feature, we can distinguish modules in complex networks, with relatively little computational time of O(NL), where N and L are the number of vertices and edges in the system, respectively. We also examine the synchronization of the modified Kuramoto model and compare it with that of the original Kuramoto model in several complex networks. Copyright c © EPLA, 2008 Complex networks have drawn considerable attention from diverse disciplines such as sociology, information science, physics, biology and so on [1]. Many complex networks in real world contain modules within them, which form in a self-organized way to achieve the efficiency functionally or regionally. Such modular systems can exhibit collective synchronized patterns within each module, not forming the global synchronization [2] as can be found in the cortex of neural network [3] or different synchronization transition behaviors depending on the patterns of inter-modular connections [4]. In this letter, we study the modular synchronization pattern generated from a modified Kuramoto equation (KE), which we call the gauge KE,