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arXiv · 1112.3644

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

C. Seshadhri·Tamara G. Kolda·Ali Pinar
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Abstract

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 Erdös-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 Erdös-Rényi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks.

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C. Seshadhri, Tamara G. Kolda, Ali Pinar. 2011-12-15. Community structure and scale-free collections of Erdös-Rényi graphs. https://doi.org/10.1103/physreve.85.056109

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