Analysis of community-detection methods based on Potts spin model in complex networks

Detection of community structures in complex networks is a common challenge in the study of complex networks. Recently, various methods have been proposed to discover community structures at different scales. Here, the multiscale methods based on Potts spin model for community detection are described and compared in the analysis of community structures of several networks. We give a critical analysis of the multiscale methods, showing a kind of limitation that the methods may suffer from when the community size difference is very broad, the breakup of (large) communities will appear before the merger of (small) communities disappears. In particular, we give the explicit expressions for the critical points of the merger and breakup of communities and derive the sufficient conditions (in the form of upper limits) that indicate when the Potts model methods suffer from the limitation. We apply the theoretical results to model networks and show that the method using the configuration null model (i.e., a random graph model as comparison that has the same degree distribution as the network under study) may not recover the full structure of the model network, whereas the method using the Erdos-Renyi null model will do so.