{"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-detection-for-binary-graphical","title":"Community detection for binary graphical models in high dimension","arxiv_id":"2411.15627","date":"2024-11-23","proceeding":null,"authors":["Julien Chevallier","Guilherme Ost"],"abstract":"Let $N$ components be partitioned into two communities, denoted ${\\cal P}_+$ and ${\\cal P}_-$, possibly of different sizes. Assume that they are connected via a directed and weighted Erd\\\"os-R\\'enyi (DWER) random graph with unknown parameter $ p \\in (0, 1).$ The weights assigned to the existing connections are of mean-field-type, scaling as $N^{-1}$. At each time \\modif{step}, we observe the state of each component: either it sends some signal to its successors (in the directed graph) or remains silent otherwise. In this paper, we show that it is possible to find the communities ${\\cal P}_+$ and ${\\cal P}_-$ based only on the activity of the $N$ components observed over $T$ time units. More specifically, we propose \\modif{ two simple methods, an aggregated method and a spectral method, whose {\\it misclassification rates} vanish as long as $T \\gg N$ (up to log terms). This condition is proved to be near-optimal in the minimax sense. Moreover, under the stronger condition $T \\gg N^2$ (up to log terms), the aggregated method is shown to achieve {\\it exact recovery} with probability tending to $1$. } Interestingly, these simple \\modif{methods} do not require any prior knowledge of the other model parameters (e.g. the edge probability $p$). The key step in our analysis is to derive an asymptotic approximation of the 1-lagged covariance matrix associated to the states of the $N$ components, as $N$ diverges. This asymptotic approximation relies on the study of the behavior of the solutions of a \\modif{Stein-type} matrix equation satisfied by the simultaneous (0-lagged) covariance matrix associated to the states of the components. This study is challenging, especially because the simultaneous covariance matrix is random since it depends on the underlying DWER random graph.","url_abs":"https://arxiv.org/abs/2411.15627v2","url_pdf":"https://arxiv.org/pdf/2411.15627v2.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-detection-for-binary-graphical","repo_url":"https://github.com/jucheval/MeanFieldGraph.jl","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}