{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T04:40:13Z","timestamp":1793162413900,"version":"4.5.0"},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2019,12,22]],"date-time":"2019-12-22T00:00:00Z","timestamp":1576972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We consider multi-layer network data where the relationships between pairs of elements are reflected in multiple modalities, and may be described by multivariate or even high-dimensional vectors. Under the multi-layer stochastic block model framework we derive consistency results for a least squares estimation of memberships. Our theorems show that, as compared to single-layer community detection, a multi-layer network provides much richer information that allows for consistent community detection from a much sparser network, with required edge density reduced by a factor of the square root of the number of layers. Moreover, the multi-layer framework can detect cohesive community structure across layers, which might be hard to detect by any single-layer or simple aggregation. Simulations and a data example are provided to support the theoretical results.<\/jats:p>","DOI":"10.1093\/biomet\/asz068","type":"journal-article","created":{"date-parts":[[2019,10,24]],"date-time":"2019-10-24T15:12:51Z","timestamp":1571929971000},"page":"61-73","source":"Crossref","is-referenced-by-count":72,"title":["Consistent community detection in multi-layer network data"],"prefix":"10.1093","volume":"107","author":[{"given":"Jing","family":"Lei","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, Carnegie Mellon University, Baker Hall 132, Pittsburgh, Pennsylvania 15213, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kehui","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Pittsburgh, 1800 Wesley W. Posvar Hall, Pittsburgh, Pennsylvania 15260, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Brian","family":"Lynch","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Pittsburgh, 1800 Wesley W. Posvar Hall, Pittsburgh, Pennsylvania 15260, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2019,12,22]]},"reference":[{"key":"2020021310532259600_B1","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1109\/TIT.2015.2490670","article-title":"Exact recovery in the stochastic block model","volume":"62","author":"Abbe,","year":"2016","journal-title":"IEEE Trans. Info. Theory"},{"key":"2020021310532259600_B2","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1214\/13-AOS1138","article-title":"Pseudo-likelihood methods for community detection in large sparse networks","volume":"41","author":"Amini,","year":"2013","journal-title":"Ann. 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