{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T02:53:04Z","timestamp":1762051984820,"version":"build-2065373602"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T00:00:00Z","timestamp":1653955200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.<\/jats:p>","DOI":"10.1017\/s0963548322000098","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T05:32:10Z","timestamp":1653975130000},"page":"1048-1069","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":2,"title":["Community detection and percolation of information in a geometric setting"],"prefix":"10.1017","volume":"31","author":[{"given":"Ronen","family":"Eldan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dan","family":"Mikulincer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hester","family":"Pieters","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,5,31]]},"reference":[{"key":"S0963548322000098_ref20","unstructured":"[20] Valdivia, E. A. (2018) Relative concentration bounds for the kernel matrix spectrum, arXiv preprint arXiv: 1812. 02108."},{"key":"S0963548322000098_ref15","first-page":"817","article-title":"Information flow on trees","volume":"13","author":"Mossel","year":"08 2003","journal-title":"Ann. Appl. Probab."},{"key":"S0963548322000098_ref7","first-page":"8581","volume-title":"Advances in Neural Information Processing Systems 31","author":"Deshpande","year":"2018"},{"key":"S0963548322000098_ref12","first-page":"2913","article-title":"Concentration of random graphs and application to community detection","volume":"3","author":"Le","year":"2018","journal-title":"Proc. Int. Cong. Math."},{"key":"S0963548322000098_ref9","unstructured":"[9] Galhotra, S. , Mazumdar, A. , Pal, S. and Saha, B.. The geometric block model. arXiv preprint arXiv: 1709. 05510, 2017."},{"key":"S0963548322000098_ref10","volume-title":"Elements of functional analysis","volume":"192","author":"Hirsch","year":"2012"},{"key":"S0963548322000098_ref5","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20633"},{"key":"S0963548322000098_ref11","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20713"},{"key":"S0963548322000098_ref2","first-page":"8724","volume-title":"Advances in Neural Information Processing Systems 32","author":"Valdivia","year":"2019"},{"key":"S0963548322000098_ref21","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asv008"},{"volume-title":"Survey: Information Flow on Trees","year":"2004","author":"Mossel","key":"S0963548322000098_ref14"},{"key":"S0963548322000098_ref16","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2013.2251927"},{"key":"S0963548322000098_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"S0963548322000098_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-011-9099-z"},{"key":"S0963548322000098_ref6","unstructured":"[6] De Castro, Y. , Lacour, C. and Ngoc, T. M. P. (2017) Adaptive estimation of nonparametric geometric graphs, arXiv preprint arXiv: 1708. 02107."},{"key":"S0963548322000098_ref17","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2013.135"},{"key":"S0963548322000098_ref18","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOS1112"},{"key":"S0963548322000098_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2015.10.010"},{"key":"S0963548322000098_ref1","doi-asserted-by":"publisher","DOI":"10.1561\/0100000067"},{"key":"S0963548322000098_ref4","first-page":"2303","article-title":"Accurate error bounds for the eigenvalues of the kernel matrix","volume":"7","author":"Braun","year":"2006","journal-title":"J. Mach. Learn. Res."},{"key":"S0963548322000098_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/998926994"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548322000098","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T04:52:11Z","timestamp":1665636731000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548322000098\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,31]]},"references-count":21,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["S0963548322000098"],"URL":"https:\/\/doi.org\/10.1017\/s0963548322000098","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2022,5,31]]},"assertion":[{"value":"\u00a9 The Author(s), 2022. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}