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 a subset of nodes that vertices in the set are well connected with other vertices in it and have relatively less connections with vertices outside of the set. Unlike the traditional research which mainly focuses on detecting and/or testing the 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 have the small community phenomenon or not, 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 small community phenomenon exists while in some other models it does not.