Multi-resolution modularity methods and their limitations in community detection

Community detection is of considerable importance for understanding the structure and function of complex networks. Recently, many multi-resolution methods have been proposed to uncover community structures of networks at different scales. Here, different multi-resolution methods are derived from modularity using self-loop assignment schemes, and then a set of multi-resolution modularity methods of this type are presented. These methods are carefully investigated by theoretical analysis of the transition points of the multi-resolution processes and experimental tests in model networks. Compared with the degree-dependent self-loop assignment, the mean-degree-dependent self-loop assignment can quicken the disconnecting of (small) communities with small vertex degrees, and can slow down the breakup of (large) communities with large vertex degrees. Moreover, we show that all these methods will encounter a limitation which is independent of the network size: large communities will break up before small communities are revealed by increasing their resolution parameters when the distribution of community sizes is very broad. Also, the tolerance of different methods against the limitation is different