Publications

Community structure and scale-free collections of Erdos-Rényi graphs

Seshadhri, C.; Kolda, Tamara G.; Pinar, Ali P.

Community structure plays a significant role in the analysis of social networks and similar graphs, yet this structure is little understood and not well captured by most models. We formally define a community to be a subgraph that is internally highly connected and has no deeper substructure. We use tools of combinatorics to show that any such community must contain a dense Erdos-Rényi (ER) subgraph. Based on mathematical arguments, we hypothesize that any graph with a heavy-tailed degree distribution and community structure must contain a scale-free collection of dense ER subgraphs. These theoretical observations corroborate well with empirical evidence. From this, we propose the Block Two-Level Erdos-Rényi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks. © 2012 American Physical Society.