Community structure and scale-free collections of Erd\"os-R\'enyi graphs

TL;DR

This paper explores the structure of communities in social networks, showing they contain dense Erd"os-Rényi subgraphs, and introduces the BTER model to better replicate real-world network properties.

Contribution

It formally defines community structure, proves the presence of dense ER subgraphs within communities, and proposes the BTER model for realistic network simulation.

Findings

Communities contain dense ER subgraphs.

Heavy-tailed degree distributions imply scale-free collections.

BTER model accurately mimics real social networks.

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 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.