{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/community-structure-and-scale-free","title":"Community structure and scale-free collections of Erdös-Rényi graphs","arxiv_id":"1112.3644","date":"2011-12-15","proceeding":null,"authors":["C. Seshadhri","Tamara G. Kolda","Ali Pinar"],"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.","url_abs":"https://arxiv.org/abs/1112.3644v1","url_pdf":"https://arxiv.org/pdf/1112.3644v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"community-structure-and-scale-free","repo_url":"https://github.com/kd383/NetworkDOS","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1112.3644","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}