{"id":"https://openalex.org/W3017750840","doi":"https://doi.org/10.48550/arxiv.2004.09780","title":"Strong Consistency, Graph Laplacians, and the Stochastic Block Model","display_name":"Strong Consistency, Graph Laplacians, and the Stochastic Block Model","publication_year":2020,"publication_date":"2020-04-21","ids":{"openalex":"https://openalex.org/W3017750840","doi":"https://doi.org/10.48550/arxiv.2004.09780","mag":"3017750840"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2004.09780","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2004.09780","pdf_url":"https://arxiv.org/pdf/2004.09780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2004.09780","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055672444","display_name":"Shaofeng Deng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Deng, Shaofeng","raw_affiliation_strings":["Univ. of California Davis"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univ. of California Davis","institution_ids":["https://openalex.org/I84218800"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047770219","display_name":"Shuyang Ling","orcid":"https://orcid.org/0000-0002-1326-3913","observed_orcids":["https://orcid.org/0000-0002-1326-3913"]},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ling, Shuyang","raw_affiliation_strings":["New York University<sup>**</sup>"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"New York University<sup>**</sup>","institution_ids":["https://openalex.org/I57206974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080565728","display_name":"Thomas Strohmer","orcid":"https://orcid.org/0000-0003-2029-3317","observed_orcids":["https://orcid.org/0000-0003-2029-3317"]},"institutions":[{"id":"https://openalex.org/I84218800","display_name":"University of California, Davis","ror":"https://ror.org/05rrcem69","country_code":"US","type":"education","lineage":["https://openalex.org/I84218800"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Strohmer, Thomas","raw_affiliation_strings":["Univ. of California Davis"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Univ. of California Davis","institution_ids":["https://openalex.org/I84218800"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.7832000255584717,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.7832000255584717,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.08250000327825546,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.01759999990463257,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/strong-consistency","display_name":"strong consistency","score":0.8460000157356262},{"id":"https://openalex.org/keywords/spectral-clustering","display_name":"spectral clustering","score":0.7870000004768372},{"id":"https://openalex.org/keywords/graph-laplacian","display_name":"graph Laplacian","score":0.9610000252723694},{"id":"https://openalex.org/keywords/stochastic-block-model","display_name":"stochastic block model","score":0.996999979019165},{"id":"https://openalex.org/keywords/community-detection","display_name":"community detection","score":0.925000011920929},{"id":"https://openalex.org/keywords/exact-recovery","display_name":"exact recovery","score":0.7179999947547913},{"id":"https://openalex.org/keywords/normalized-laplacian-matrix","display_name":"normalized Laplacian","score":0.7089999914169312},{"id":"https://openalex.org/keywords/information-theoretic-limits","display_name":"information-theoretic limits","score":0.8510000109672546}],"concepts":[{"id":"https://openalex.org/C2779982251","wikidata":"https://www.wikidata.org/wiki/Q25053762","display_name":"Stochastic block model","level":3,"score":0.9324919581413269},{"id":"https://openalex.org/C105611402","wikidata":"https://www.wikidata.org/wiki/Q2976589","display_name":"Spectral clustering","level":3,"score":0.8565608859062195},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.8288166522979736},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.7663547992706299},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.7140359878540039},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6307082176208496},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6155346035957336},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.5337292551994324},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5218902230262756},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.47351861000061035},{"id":"https://openalex.org/C2993597952","wikidata":"https://www.wikidata.org/wiki/Q108123644","display_name":"Block model","level":2,"score":0.46659374237060547},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.466114342212677},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42620474100112915},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.4179074764251709},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40316957235336304},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34858667850494385},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11664921045303345},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.09987619519233704},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0887298583984375},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.08088886737823486},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07748749852180481},{"id":"https://openalex.org/C16674752","wikidata":"https://www.wikidata.org/wiki/Q1370637","display_name":"Mining engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:2004.09780","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2004.09780","pdf_url":"https://arxiv.org/pdf/2004.09780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:3017750840","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2004.09780.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2004.09780","doi":"https://doi.org/10.48550/arxiv.2004.09780","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2004.09780","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2004.09780","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2004.09780","pdf_url":"https://arxiv.org/pdf/2004.09780","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W264913434","https://openalex.org/W1510081914","https://openalex.org/W1526954821","https://openalex.org/W1593316675","https://openalex.org/W1605711022","https://openalex.org/W1855425795","https://openalex.org/W1914749871","https://openalex.org/W1970377488","https://openalex.org/W1971421925","https://openalex.org/W1974967023","https://openalex.org/W1989042829","https://openalex.org/W2004531067","https://openalex.org/W2009034807","https://openalex.org/W2021145763","https://openalex.org/W2023348178","https://openalex.org/W2023655578","https://openalex.org/W2064274783","https://openalex.org/W2092164538","https://openalex.org/W2102907934","https://openalex.org/W2121947440","https://openalex.org/W2127048411","https://openalex.org/W2132914434","https://openalex.org/W2148606196","https://openalex.org/W2156718197","https://openalex.org/W2165874743","https://openalex.org/W2216459995","https://openalex.org/W2290127356","https://openalex.org/W2302206641","https://openalex.org/W2408125176","https://openalex.org/W2559839022","https://openalex.org/W2567866584","https://openalex.org/W2598300585","https://openalex.org/W2765311429","https://openalex.org/W2768927505","https://openalex.org/W2787248994","https://openalex.org/W2791984915","https://openalex.org/W2963037546","https://openalex.org/W2963264680","https://openalex.org/W3024319689","https://openalex.org/W3043617248","https://openalex.org/W3100144903","https://openalex.org/W3104227803"],"related_works":["https://openalex.org/W2765311429","https://openalex.org/W3100144903","https://openalex.org/W7051724775","https://openalex.org/W2969605534","https://openalex.org/W2722895473","https://openalex.org/W2051638434","https://openalex.org/W2806276374","https://openalex.org/W4400449646","https://openalex.org/W1967184357","https://openalex.org/W7099741048"],"abstract_inverted_index":{"Spectral":[0],"clustering":[1,12,24,46,102],"has":[2],"become":[3],"one":[4],"of":[5,20,60,75,79,109],"the":[6,18,26,31,40,48,57,61,76,81,84,89,94,110,118],"most":[7],"popular":[8],"algorithms":[9],"in":[10],"data":[11],"and":[13,83],"community":[14,112],"detection.":[15],"We":[16,98],"study":[17],"performance":[19],"classical":[21],"two-step":[22],"spectral":[23,45,101],"via":[25,47],"graph":[27,49],"Laplacian":[28,50,86],"to":[29,38,52,105],"learn":[30],"stochastic":[32,95],"block":[33,96],"model.":[34,97],"Our":[35,65],"aim":[36],"is":[37,44,103],"answer":[39],"following":[41],"question:":[42],"when":[43],"able":[51,104],"achieve":[53,106],"strong":[54],"consistency,":[55],"i.e.,":[56],"exact":[58,107],"recovery":[59,108],"underlying":[62],"hidden":[63],"communities?":[64],"work":[66],"provides":[67],"an":[68],"entrywise":[69],"analysis":[70],"(an":[71],"$\\ell_{\\infty}$-norm":[72],"perturbation":[73],"bound)":[74],"Fielder":[77],"eigenvector":[78],"both":[80],"unnormalized":[82],"normalized":[85],"associated":[87],"with":[88],"adjacency":[90],"matrix":[91],"sampled":[92],"from":[93],"prove":[99],"that":[100,116],"planted":[111],"structure":[113],"under":[114],"conditions":[115],"match":[117],"information-theoretic":[119],"limits.":[120]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":2}],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2020-05-01T00:00:00"}