Modular synchronization in complex networks with a gauge Kuramoto model

We modify the Kuramoto equation(KE) by introducing a gauge term which is a function of link betweenness centrality(BC). The gauge term induces the phase difference from 0 to π between two nodes that belong to different modules. Therefore, a synchronization occurs in each module individually even though the whole network is not synchronized globally. By measuring the phase similarity of all pairs of connected nodes, we can detect the modular structure of complex networks. This algorithm requires relatively little computational time O(NL) for network with N nodes and L links.