{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T14:29:32Z","timestamp":1762352972321,"version":"3.40.5"},"reference-count":55,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-18-1-0244"],"award-info":[{"award-number":["W911NF-18-1-0244"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006510","name":"Duke University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100006510","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000062","name":"National Institute of Diabetes and Digestive and Kidney Diseases","doi-asserted-by":"publisher","award":["R01DK111930"],"award-info":[{"award-number":["R01DK111930"]}],"id":[{"id":"10.13039\/100000062","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["T32CA201159","T32GM67553"],"award-info":[{"award-number":["T32CA201159","T32GM67553"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000913","name":"James S. McDonnell Foundation","doi-asserted-by":"publisher","award":["220020315"],"award-info":[{"award-number":["220020315"]}],"id":[{"id":"10.13039\/100000913","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Journal on Mathematics of Data Science"],"published-print":{"date-parts":[[2020,1]]},"DOI":"10.1137\/19m1279812","type":"journal-article","created":{"date-parts":[[2020,9,28]],"date-time":"2020-09-28T19:04:43Z","timestamp":1601319883000},"page":"872-900","source":"Crossref","is-referenced-by-count":3,"title":["Multilayer Modularity Belief Propagation to Assess Detectability of Community Structure"],"prefix":"10.1137","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7159-1990","authenticated-orcid":true,"given":"William H.","family":"Weir","sequence":"first","affiliation":[]},{"given":"Benjamin","family":"Walker","sequence":"additional","affiliation":[]},{"given":"Lenka","family":"Zdeborov\u00e1","sequence":"additional","affiliation":[]},{"given":"Peter J.","family":"Mucha","sequence":"additional","affiliation":[]}],"member":"351","published-online":{"date-parts":[[2020,9,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","first-page":"053039","DOI":"10.1088\/1367-2630\/10\/5\/053039","volume":"10","author":"Arenas A.","year":"2008","journal-title":"New J. Phys."},{"key":"atypb2","doi-asserted-by":"crossref","first-page":"066118","DOI":"10.1103\/PhysRevE.85.066118","volume":"85","author":"Bagrow J. P.","year":"2012","journal-title":"Phys. Rev. E"},{"key":"atypb3","doi-asserted-by":"crossref","first-page":"023100","DOI":"10.1103\/PhysRevResearch.2.023100","volume":"2","author":"Bazzi M.","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/15M1009615"},{"key":"atypb5","doi-asserted-by":"crossref","first-page":"e112821","DOI":"10.1371\/journal.pone.0112821","volume":"9","author":"Bennett L.","year":"2014","journal-title":"PLOS ONE"},{"key":"atypb6","doi-asserted-by":"crossref","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","volume":"2008","author":"Blondel V. D.","year":"2008","journal-title":"J. Stat. Mech. Theory Experiment"},{"key":"atypb7","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1109\/TKDE.2007.190689","volume":"20","author":"Brandes U.","year":"2008","journal-title":"IEEE Trans. Knowledge Data Engrg."},{"key":"atypb8","first-page":"435","author":"de Montgolfier F.","year":"2011","journal-title":"Berlin"},{"key":"atypb9","doi-asserted-by":"crossref","first-page":"066106","DOI":"10.1103\/PhysRevE.84.066106","volume":"84","author":"Decelle A.","year":"2011","journal-title":"Phys. Rev. E"},{"key":"atypb10","doi-asserted-by":"crossref","first-page":"065701","DOI":"10.1103\/PhysRevLett.107.065701","volume":"107","author":"Decelle A.","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"atypb11","doi-asserted-by":"crossref","first-page":"P12037","DOI":"10.1088\/1742-5468\/2010\/12\/P12037","volume":"2010","author":"Evans T. S.","year":"2010","journal-title":"J. Stat. Mech. Theory Experiment"},{"key":"atypb12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.physrep.2009.11.002","volume":"486","author":"Fortunato S.","year":"2010","journal-title":"Phys. Rep."},{"key":"atypb13","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1073\/pnas.0605965104","volume":"104","author":"Fortunato S.","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"atypb14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.09.002","volume":"659","author":"Fortunato S.","year":"2016","journal-title":"Phys. Rep."},{"key":"atypb15","first-page":"031005","volume":"6","author":"Ghasemian A.","year":"2016","journal-title":"Phys. Rev. X"},{"key":"atypb16","doi-asserted-by":"crossref","first-page":"7821","DOI":"10.1073\/pnas.122653799","volume":"99","author":"Girvan M.","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"atypb18","doi-asserted-by":"crossref","first-page":"035102","DOI":"10.1103\/PhysRevE.74.035102","volume":"74","author":"Hastings M. B.","year":"2006","journal-title":"Phys. Rev. E"},{"key":"atypb19","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1109\/TFUZZ.2013.2245135","volume":"21","author":"Havens T. C.","year":"2013","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"atypb20","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","volume":"5","author":"Holland P. W.","year":"1983","journal-title":"Social Networks"},{"key":"atypb21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1371\/journal.pone.0098679","volume":"9","author":"Jacomy M.","year":"2014","journal-title":"PLOS ONE"},{"key":"atypb25","doi-asserted-by":"crossref","first-page":"016107","DOI":"10.1103\/PhysRevE.83.016107","volume":"83","author":"Karrer B.","year":"2011","journal-title":"Phys. Rev. E"},{"key":"atypb26","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1093\/comnet\/cnu016","volume":"2","author":"Kivel\u00e4 M.","year":"2014","journal-title":"J. Complex Networks"},{"key":"atypb27","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1002\/nav.3800020109","volume":"2","author":"Kuhn H. W.","year":"1955","journal-title":"Naval Res. Logist. Quart."},{"key":"atypb28","doi-asserted-by":"crossref","first-page":"053301","DOI":"10.1088\/1742-5468\/2016\/05\/053301","volume":"2016","author":"Lai D.","year":"2016","journal-title":"J. Stat. Mech. Theory Experiment"},{"key":"atypb30","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1109\/TNSE.2015.2391998","volume":"1","author":"Lambiotte R.","year":"2014","journal-title":"IEEE Trans. Network Sci. Engrg."},{"key":"atypb31","doi-asserted-by":"crossref","first-page":"046110","DOI":"10.1103\/PhysRevE.78.046110","volume":"78","author":"Lancichinetti A.","year":"2008","journal-title":"Phys. Rev. E"},{"key":"atypb33","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1140\/epjb\/e2010-00290-3","volume":"77","author":"Liu J.","year":"2010","journal-title":"European Phys. J. B"},{"key":"atypb35","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1017\/nws.2012.3","volume":"1","author":"Moody J.","year":"2013","journal-title":"Network Sci."},{"key":"atypb36","first-page":"356","author":"Mossel E.","year":"2014","journal-title":"China"},{"key":"atypb37","doi-asserted-by":"crossref","first-page":"041108","DOI":"10.1063\/1.3518696","volume":"20","author":"Mucha P. J.","year":"2010","journal-title":"Chaos"},{"key":"atypb38","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1126\/science.1184819","volume":"328","author":"Mucha P. J.","year":"2010","journal-title":"Science"},{"key":"atypb39","doi-asserted-by":"crossref","first-page":"188701","DOI":"10.1103\/PhysRevLett.108.188701","volume":"108","author":"Nadakuditi R. R.","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"atypb40","doi-asserted-by":"crossref","first-page":"026113","DOI":"10.1103\/PhysRevE.69.026113","volume":"69","author":"Newman M.","year":"2004","journal-title":"Phys. Rev. E"},{"key":"atypb41","doi-asserted-by":"crossref","first-page":"056131","DOI":"10.1103\/PhysRevE.70.056131","volume":"70","author":"M. E.","year":"2004","journal-title":"Phys. Rev. E"},{"key":"atypb42","first-page":"052315","volume":"94","author":"M. E.","year":"2016","journal-title":"Phys. Rev. E"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/18M1231304"},{"key":"atypb44","first-page":"341","author":"Papadopoulos A.","year":"2015","journal-title":"Cham"},{"key":"atypb46","doi-asserted-by":"crossref","first-page":"e1602548","DOI":"10.1126\/sciadv.1602548","volume":"3","author":"Peel L.","year":"2017","journal-title":"Sci. Adv."},{"key":"atypb47","doi-asserted-by":"crossref","first-page":"012317","DOI":"10.1103\/PhysRevE.95.012317","volume":"95","author":"Peixoto T. P.","year":"2017","journal-title":"Phys. Rev. E"},{"key":"atypb48","first-page":"1082","volume":"56","author":"Porter M. A.","year":"2009","journal-title":"Notices Amer. Math. Soc."},{"key":"atypb49","doi-asserted-by":"crossref","first-page":"016110","DOI":"10.1103\/PhysRevE.74.016110","volume":"74","author":"Reichardt J.","year":"2006","journal-title":"Phys. Rev. E"},{"key":"atypb50","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1073\/pnas.0706851105","volume":"105","author":"Rosvall M.","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"atypb51","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/s41109-017-0023-6","volume":"2","author":"Schaub M. T.","year":"2017","journal-title":"Appl. Network Sci."},{"key":"atypb52","doi-asserted-by":"crossref","first-page":"042804","DOI":"10.1103\/PhysRevE.92.042804","volume":"92","author":"Sch\u00fclke C.","year":"2015","journal-title":"Phys. Rev. E"},{"key":"atypb54","doi-asserted-by":"crossref","first-page":"033405","DOI":"10.1088\/1742-5468\/aaa8f5","volume":"2018","author":"Shi C.","year":"2018","journal-title":"J. Stat. Mech. Theory Experiment"},{"key":"atypb55","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/TNSE.2016.2537545","volume":"3","author":"Stanley N.","year":"2016","journal-title":"IEEE Trans. Network Sci. Engrg."},{"key":"atypb56","doi-asserted-by":"crossref","first-page":"228301","DOI":"10.1103\/PhysRevLett.116.228301","volume":"116","author":"Taylor D.","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"atypb57","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1038\/srep02930","volume":"3","author":"Traag V. A.","year":"2013","journal-title":"Sci. Rep."},{"key":"atypb58","first-page":"1073","author":"Vinh N. X.","year":"2009","journal-title":"Canada"},{"key":"atypb59","first-page":"2837","volume":"11","author":"Vinh N. X.","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"atypb61","doi-asserted-by":"crossref","first-page":"93","DOI":"10.3390\/a10030093","volume":"10","author":"Weir W. H.","year":"2017","journal-title":"Algorithms (Basel)"},{"key":"atypb64","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1080\/00018732.2016.1211393","volume":"65","author":"Zdeborov\u00e1 L.","year":"2016","journal-title":"Adv. Phys."},{"key":"atypb65","doi-asserted-by":"crossref","first-page":"P12021","DOI":"10.1088\/1742-5468\/2012\/12\/P12021","volume":"2012","author":"Zhang P.","year":"2012","journal-title":"J. Stat. Mech. Theory Experiment"},{"key":"atypb66","doi-asserted-by":"crossref","first-page":"18144","DOI":"10.1073\/pnas.1409770111","volume":"111","author":"Zhang P.","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"atypb67","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.physa.2006.07.023","volume":"374","author":"Zhang S.","year":"2007","journal-title":"Phys. A"}],"container-title":["SIAM Journal on Mathematics of Data Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1279812","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,30]],"date-time":"2020-09-30T19:25:31Z","timestamp":1601493931000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1279812"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":55,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1279812"],"URL":"https:\/\/doi.org\/10.1137\/19m1279812","relation":{},"ISSN":["2577-0187"],"issn-type":[{"type":"electronic","value":"2577-0187"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}