Detecting the Structure of Social Networks Using (alpha, beta)-Communities under Grant 2007CB807900 and 2007CB807901.

An (alpha, beta)-community is a subset of vertices C with each vertex in C connected to at least beta vertices of C (self-loops counted) and each vertex outside of C connected to at most a vertices of C (alpha, beta) [ 9]. In this paper, we present a heuristic (alpha, beta)-Community algorithm, which in practice successfully finds (alpha, beta)-communities of a given size. The structure of (alpha, beta)-communities in several large-scale social graphs is explored, and a surprising core structure is discovered by taking the intersection of a group of massively overlapping (alpha, beta)-communities. For large community size k, the (alpha, beta)-communities are well clustered into a small number of disjoint cores, and there are no isolated (alpha, beta)-communities scattered between these densely-clustered cores. The (alpha, beta)-communities from the same group have significant overlap among them, and those from distinct groups have extremely small pairwise resemblance. The number of cores decreases as k increases, and there are no bridges of intermediate (alpha, beta)-communities connecting one core to another. The cores obtained for a smaller k either disappear or merge into the cores obtained for a larger k. Further, similar experiments on random graph models demonstrate that the core structure displayed in various social graphs is due to the underlying social structure of these real-world networks, rather than due to high-degree vertices or a particular degree distribution.