On Modularity of Social Network Communities: The Spectral Characterization

The term of social network communities refers to groups of individuals within which social interactions are intense and between which they are weak. A social network community mining problem (SNCMP) can be stated as the problem of finding all such communities from a given social network. A wide variety of applications can be formulated into SNCMPs, ranging from Web intelligence to social intelligence. So far, many algorithms addressing the SNCMP have been developed; most of them are either optimization or heuristic based methods. Different from all existing work, this paper explores the notion of a social network community and its intrinsic properties, drawing on the dynamics of a stochastic model naturally introduced. In particular, it uncovers an interesting connection between the hierarchical community structure of a network and the metastability of a Markov process constructed upon it. A lot of critical topological information regarding to communities hidden in networks can be inferred from the derived spectral signatures of such networks, without actually clustering them with any particular algorithms. Based upon the above connection, we can obtain a framework for characterizing and analyzing social network communities.