{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T01:34:30Z","timestamp":1790991270580,"version":"4.1.0"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T00:00:00Z","timestamp":1648598400000},"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":[[2022,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Network data are prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.<\/jats:p>","DOI":"10.1111\/rssb.12505","type":"journal-article","created":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T13:46:26Z","timestamp":1648647986000},"page":"630-653","source":"Crossref","is-referenced-by-count":35,"title":["Simple: Statistical Inference on Membership Profiles in Large Networks"],"prefix":"10.1093","volume":"84","author":[{"given":"Jianqing","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Operations Research and Financial Engineering , , Princeton, New Jersey , USA"},{"name":"Princeton University , , Princeton, New Jersey , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingying","family":"Fan","sequence":"additional","affiliation":[{"name":"Data Sciences and Operations Department , , , Los Angeles, California , USA"},{"name":"Marshall School of Business , , , Los Angeles, California , USA"},{"name":"University of Southern California , , , Los Angeles, California , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao","family":"Han","sequence":"additional","affiliation":[{"name":"International Institute of Finance , , , Hefei , China"},{"name":"Department of Statistics and Finance , , , Hefei , China"},{"name":"University of Science and Technology of China , , , Hefei , China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinchi","family":"Lv","sequence":"additional","affiliation":[{"name":"Data Sciences and Operations Department , , , Los Angeles, California , USA"},{"name":"Marshall School of Business , , , Los Angeles, California , USA"},{"name":"University of Southern California , , , Los Angeles, California , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,3,30]]},"reference":[{"key":"2023022408355568500_","first-page":"177:1","article-title":"Community detection and stochastic block models: recent developments","volume":"18","author":"Abbe","year":"2017","journal-title":"Journal of Machine Learning Research"},{"key":"2023022408355568500_","article-title":"Entrywise eigenvector analysis of random matrices with low expected rank","volume-title":"arXiv preprint arXiv:1709.09565","author":"Abbe","year":"2017"},{"key":"2023022408355568500_","first-page":"1981","article-title":"Mixed membership stochastic blockmodels","volume":"9","author":"Airoldi","year":"2008","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"940","DOI":"10.1214\/14-AOS1208","article-title":"Community detection in dense random networks","volume":"42","author":"Arias-Castro","year":"2014","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/rssb.12117","article-title":"Hypothesis testing for automated community detection in networks","volume":"78","author":"Bickel","year":"2016","journal-title":"Journal of the Royal Statistical Society Series B"},{"key":"2023022408355568500_","article-title":"Community detection in partial correlation network models","volume-title":"Manuscript","author":"Brownlees","year":"2019"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/01621459.2016.1246365","article-title":"Network cross-validation for determining the number of communities in network data","volume":"113","author":"Chen","year":"2018","journal-title":"Journal of the American Statistical Association"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s11222-007-9046-7","article-title":"A mixture model for random graphs","volume":"18","author":"Daudin","year":"2008","journal-title":"Statistics and Computing"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.jeconom.2008.09.017","article-title":"High dimensional covariance matrix estimation using a factor model","volume":"147","author":"Fan","year":"2008","journal-title":"Journal of Econometrics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1111\/rssb.12016","article-title":"Large covariance estimation by thresholding principal orthogonal complements (with discussion)","volume":"75","author":"Fan","year":"2013","journal-title":"Journal of the Royal Statistical Society Series B"},{"key":"2023022408355568500_","article-title":"Asymptotic theory of eigenvectors for random matrices with diverging spikes","author":"Fan","year":"2020","journal-title":"Journal of the American Statistical Association"},{"issue":"5","key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1214\/17-AOS1615","article-title":"Community detection in degree- corrected block models","volume":"46","author":"Gao","year":"2018","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1561\/2200000005","article-title":"A survey of statistical network models","volume":"2","author":"Goldenberg","year":"2010","journal-title":"Foundations and Trends in Machine Learning"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","article-title":"Stochastic blockmodels: first steps","volume":"5","author":"Holland","year":"1983","journal-title":"Social Networks"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1214\/14-AOS1265","article-title":"Fast community detection by SCORE","volume":"43","author":"Jin","year":"2015","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","article-title":"A sharp lower bound for mixed-membership estimation","volume-title":"arXiv preprint arXiv:1709.05603","author":"Jin","year":"2017"},{"key":"2023022408355568500_","article-title":"Estimating network memberships by simplex vertex hunting","volume-title":"arXiv preprint arXiv:1708.07852","author":"Jin","year":"2017"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"16107","DOI":"10.1103\/PhysRevE.83.016107","article-title":"Stochastic blockmodels and community structure in networks","volume":"83","author":"Karrer","year":"2011","journal-title":"Physical Review E"},{"key":"2023022408355568500_","first-page":"48","article-title":"Finding mixed-memberships in social net-works","author":"Koutsourelakis","year":"2008","journal-title":"AAAI Spring Symposium: Social Information Processing"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1177\/1471082X1001200105","article-title":"Variational bayesian inference and complexity control for stochastic block models","volume":"12","author":"Latouche","year":"2012","journal-title":"Statistical Modelling"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1214\/15-AOS1370","article-title":"A goodness-of-fit test for stochastic block models","volume":"44","author":"Lei","year":"2016","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1214\/14-AOS1274","article-title":"Consistency of spectral clustering in stochastic block models","volume":"43","author":"Lei","year":"2015","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","first-page":"1019","article-title":"The link-prediction problem for social networks","volume":"58","author":"Liben-Nowell","year":"2007","journal-title":"Journal of the Association for Information Science and Technology"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11222-007-9033-z","article-title":"A tutorial on spectral clustering","volume":"17","author":"von Luxburg","year":"2007","journal-title":"Statistics and Computing"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"28003","DOI":"10.1209\/0295-5075\/103\/28003","article-title":"Community detection and graph partitioning","volume":"103","author":"Newman","year":"2013","journal-title":"EPL (Europhysics Letters)"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"42822","DOI":"10.1103\/PhysRevE.88.042822","article-title":"Spectral methods for community detection and graph partitioning","volume":"88","author":"Newman","year":"2013","journal-title":"Physical Review E"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"88701","DOI":"10.1103\/PhysRevLett.115.088701","article-title":"Generalized communities in networks","volume":"115","author":"Newman","year":"2015","journal-title":"Physical Review Letters"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1214\/11-AOS887","article-title":"Spectral clustering and the high-dimensional stochastic blockmodel","volume":"39","author":"Rohe","year":"2011","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1080\/10618600.2015.1096790","article-title":"How many communities are there","volume":"26,","author":"Saldana","year":"2017","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"6","key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"3465","DOI":"10.1214\/14-AAP1080","article-title":"Community detection in sparse random networks","volume":"25","author":"Verzelen","year":"2015","journal-title":"Annals of Applied Probability"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1214\/16-AOS1457","article-title":"Likelihood-based model selection for stochastic block models","volume":"45","author":"Wang","year":"2017","journal-title":"The Annals of Statistics"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1080\/01621459.1987.10478385","article-title":"Stochastic blockmodels for directed graphs","volume":"82","author":"Wang","year":"1987","journal-title":"Journal of the American Statistical Association"},{"key":"2023022408355568500_","article-title":"Link prediction for egocentrically sampled networks","volume-title":"arXiv preprint arXiv:1803.040845","author":"Wu","year":"2018"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"18144","DOI":"10.1073\/pnas.1409770111","article-title":"Scalable detection of statistically significant communities and hierarchies, using message passing for modularity","volume":"111","author":"Zhang","year":"2014","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"2","key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1137\/19M1272238","article-title":"Detecting overlapping communities in networks using spectral methods","volume":"2","author":"Zhang","year":"2020","journal-title":"SIAM Journal on Mathematics of Data Science"},{"key":"2023022408355568500_","doi-asserted-by":"crossref","first-page":"2266","DOI":"10.1214\/12-AOS1036","article-title":"Consistency of community detection in networks under degree-corrected stochastic block models","volume":"40","author":"Zhao","year":"2012","journal-title":"The Annals of Statistics"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/rssb.12505","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/rssb.12505","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/84\/2\/630\/49321201\/jrsssb_84_2_630.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/84\/2\/630\/49321201\/jrsssb_84_2_630.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,19]],"date-time":"2023-11-19T12:17:02Z","timestamp":1700396222000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/84\/2\/630\/7056066"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,30]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,3,30]]},"published-print":{"date-parts":[[2022,4,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12505","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,4]]},"published":{"date-parts":[[2022,3,30]]}}}