Community structures in classical network models

Communities (or clusters) are ubiquitous in real-world networks. Researchers from different fields have proposed many definitions of communities, which are usually thought of as a subset of nodes whose vertices are well connected with other vertices in the set and have relatively fewer connections with vertices outside the set. In contrast to traditional research that focuses mainly on detecting and/or testing such clusters, we propose a new definition of community and a novel way to study community structure, with which we are able to investigate mathematical network models to test whether they exhibit the small-community phenomenon, i.e., whether every vertex in the network belongs to some small community. We examine various models and establish both positive and negative results: we show that in some models, the small-community phenomenon exists, while in some other models, it does not.