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

Details Community structure and scale-free collections of Erdös-Rényi graphs Community structure and scale-free collections of Erdös-Rényi graphs

2011-12-15

arxiv.org/abs/1112.3644v1

Computer Science

Social and Information Networks

Scientific paper

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 Erd\"os-R\'enyi (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 Erd\"os-R\'enyi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks.

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