Community Structure in Large Complex Networks

In this paper, we establish the definition of community fundamentally different from what was commonly accepted in previous studies, where communities were typically assumed to be densely connected internally but sparsely connected to the rest of the network. A community should be considered as a densely connected subset in which the probability of an edge between two randomly-picked vertices is higher than average. Moreover, a community should also be well connected to the remaining network, that is, the number of edges connecting a community to the rest of the graph should be significant. In order to identify a well-defined community, we provide rigorous definitions of two relevant terms: “whiskers” and the “core”. Whiskers correspond to subsets of vertices that are barely connected to the rest of the network, while the core exclusively contains the type of community we are interested in. We have proven that detecting whiskers, or equivalently, extracting the core, is an NP-complete problem for weighted graphs. Then, three heuristic algorithms are proposed for finding an approximate core and are evaluated for their performance on large networks, which reveals the common existence of the core structure in both random and real-world graphs. Further, well-defined communities can be extracted from the core using a number of techniques, and the experimental results not only justify our intuitive notion of community, but also demonstrate the existence of large-scale communities in various complex networks.