Detection of overlapping and hierarchical modular structure in networks using distance analysis between neighbors of each node

Hierarchical modular structure is an inherent characteristic of real networks, including social and biological networks. Although extensive mathematical research has been performed on community detection, the detection of hierarchical communities with overlaps is still in the development stage. In this paper, we propose a method for detecting overlapping and hierarchical modular structure, assuming that the modular structure of a network is generated by rare connections between disconnected clusters that usually evolve through the formation of local triangular relations. Under this assumption, we infer that nodes with a large harmonic mean distance between neighbors (H-mean DN) tend to be core nodes of bridging structure between different clusters. In our scheme, we intend the network to fall apart completely by iteratively removing the core nodes and the links in their neighborhoods, so we perform the agglomerative hierarchical clustering by utilizing the degree of overlaps. We confirm that the method is effective at detecting modular structures of benchmark networks, and that the overlapped regions contain critical information about inter-cluster links. Application to model networks also shows that our method provides significant information about the evolutionary processes of networks that are consistent with our assumptions. In particular, the highly hierarchical structures detected in some social networks suggest a complex process in which numerous disconnected clusters merge.