Characters and patterns of communities in networks

A community can be seen as a group of vertices with strong cohesion among themselves and weak cohesion between each other. Community structure is one of the most remarkable features of many complex networks. There are various kinds of algorithms for detecting communities. However it is widely open for the question: what can we do with the communities? In this paper, we propose some new notions to characterize and analyze the communities. The new notions are general characters of the communities or local structures of networks. At first, we introduce the notions of internal dominating set and external dominating set of a community. We show that most communities in real networks have a small internal dominating set and a small external dominating set, and that the internal dominating set of a community keeps much of the information of the community. Secondly, based on the notions of the internal dominating set and the external dominating set, we define an internal slope (ISlope, for short) and an external slope (ESlope, for short) to measure the internal heterogeneity and external heterogeneity of a community respectively. We show that the internal slope (ISlope) of a community largely determines the structure of the community, that most communities in real networks are heterogeneous, meaning that most of the communities have a core/periphery structure, and that both ISlopes and ESlopes (reflecting the structure of communities) of all the communities of a network approximately follow a normal distribution. Therefore typical values of both ISolpes and ESoples of all the communities of a given network are in a narrow interval, and there is only a small number of communities having ISlopes or ESlopes out of the range of typical values of the ISlopes and ESlopes of the network. Finally, we show that all the communities of the real networks we studied, have a three degree separation phenomenon, that is, the average distance of communities is approximately 3, implying a general property of true communities for many real networks, and that good community finding algorithms find communities that amplify clustering coefficients of the networks, for many real networks.