  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 $\pi$ 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 $\mathcal{O}(NL)$, 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.
