Multipolarization versus unification in community networks

Community structure and core-periphery structure are two natural properties of complex networks. Both structures have been studied separately for decades. However, few researchers focus on the combination of these two important structures in complex networks. In this paper, we explore the core-periphery structures of communities in complex networks especially community networks, more precisely, we propose a linear algorithm to divide each community into a densely interconnected core and a periphery where the nodes are rarely linked to each other. Based on core-periphery structures, we perform quantitative analysis of the edges between different communities and find two relationships of two communities in real networks: unitive and multipolar. Communities are called unitive if edges between different cores are more than edges between different peripheries. Otherwise, communities are called multipolar. Furthermore, we propose a random model called Generalized Girvan-Newman(GGN) model, which can generate community networks where communities are either unitive or multipolar. The model sheds some new light on community formation and core-periphery structures in complex systems.