{"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/consistency-of-modularity-clustering-on","title":"Consistency of modularity clustering on random geometric graphs","arxiv_id":"1604.03993","date":"2016-04-13","proceeding":null,"authors":["Erik Davis","Sunder Sethuraman"],"abstract":"We consider a large class of random geometric graphs constructed from samples $\\mathcal{X}_n = \\{X_1,X_2,\\ldots,X_n\\}$ of independent, identically distributed observations of an underlying probability measure $\\nu$ on a bounded domain $D\\subset \\mathbb{R}^d$. The popular `modularity' clustering method specifies a partition $\\mathcal{U}_n$ of the set $\\mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $\\nu$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $\\mathcal{U}_n$ converge in a certain sense to a continuum partition $\\mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvin's shape optimization problem.","url_abs":"http://arxiv.org/abs/1604.03993v1","url_pdf":"http://arxiv.org/pdf/1604.03993v1.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":"consistency-of-modularity-clustering-on","repo_url":"https://github.com/Exa-Graph/miniVite","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"consistency-of-modularity-clustering-on","repo_url":"https://github.com/Exa-Graph/vite","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}