Extracting the Core Structure of Social Networks Using (α, β)-Communities

An (α, β)-community is a connected subgraph C with each vertex in C connected to at least β vertices of C (self-loops counted) and each vertex outside of C connected to at most α vertices of C (α<β). In this paper, we present a heuristic algorithm that in practice successfully finds a fundamental community structure. We also explore the structure of (α, β)-communities in various social networks. (α, β)-communities are well clustered into a small number of disjoint groups, and there are no isolated (α, β)-communities scattered between these groups. Two (α, β)-communities in the same group have significant overlap, while those in different groups have extremely small resemblance. A surprising core structure is discovered by taking the intersection of each group of massively overlapping (α, β)-communities. Further, similar experiments on random graphs demonstrate that the core structure found in many social networks is due to their underlying social structure, rather than to high-degree vertices or a particular degree distribution.