{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T06:20:40Z","timestamp":1776752440678,"version":"3.51.2"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2020,5,27]],"date-time":"2020-05-27T00:00:00Z","timestamp":1590537600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,5,27]],"date-time":"2020-05-27T00:00:00Z","timestamp":1590537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000761","name":"Imperial College London","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000761","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Spectral embedding of adjacency or Laplacian matrices of undirected graphs is a common technique for representing a network in a lower dimensional latent space, with optimal theoretical guarantees. The embedding can be used to estimate the community structure of the network, with strong consistency results in the stochastic blockmodel framework. One of the main practical limitations of standard algorithms for community detection from spectral embeddings is that the number of communities and the latent dimension of the embedding must be specified in advance. In this article, a novel Bayesian model for simultaneous and automatic selection of the appropriate dimension of the latent space and the number of blocks is proposed. Extensions to directed and bipartite graphs are discussed. The model is tested on simulated and real world network data, showing promising performance for recovering latent community structure.<\/jats:p>","DOI":"10.1007\/s11222-020-09946-6","type":"journal-article","created":{"date-parts":[[2020,5,27]],"date-time":"2020-05-27T00:02:36Z","timestamp":1590537756000},"page":"1291-1307","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Bayesian estimation of the latent dimension and communities in stochastic blockmodels"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4571-6681","authenticated-orcid":false,"given":"Francesco Sanna","family":"Passino","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8767-0810","authenticated-orcid":false,"given":"Nicholas A.","family":"Heard","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,5,27]]},"reference":[{"issue":"4","key":"9946_CR1","doi-asserted-by":"crossref","first-page":"2097","DOI":"10.1214\/13-AOS1138","volume":"41","author":"AA Amini","year":"2013","unstructured":"Amini, A.A., Chen, A., Bickel, P.J., Levina, E.: Pseudo-likelihood methods for community detection in large sparse networks. Ann. Stat. 41(4), 2097\u20132122 (2013)","journal-title":"Ann. Stat."},{"issue":"1","key":"9946_CR2","first-page":"8393","volume":"18","author":"A Athreya","year":"2017","unstructured":"Athreya, A., Fishkind, D.E., Tang, M., Priebe, C.E., Park, Y., Vogelstein, J.T., Levin, K., Lyzinski, V., Qin, Y.: Statistical inference on random dot product graphs: a survey. J. Mach. Learn. Res. 18(1), 8393\u20138484 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"9946_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13171-015-0071-x","volume":"78","author":"A Athreya","year":"2016","unstructured":"Athreya, A., Priebe, C.E., Tang, M., Lyzinski, V., Marchette, D.J., Sussman, D.L.: A limit theorem for scaled eigenvectors of random dot product graphs. Sankhya A 78(1), 1\u201318 (2016)","journal-title":"Sankhya A"},{"issue":"50","key":"9946_CR4","doi-asserted-by":"crossref","first-page":"21068","DOI":"10.1073\/pnas.0907096106","volume":"106","author":"PJ Bickel","year":"2009","unstructured":"Bickel, P.J., Chen, A.: A nonparametric view of network models and Newman\u2013Girvan and other modularities. Proc. Natl. Acad. Sci. 106(50), 21068\u201321073 (2009)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"1","key":"9946_CR5","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1111\/rssb.12117","volume":"78","author":"PJ Bickel","year":"2016","unstructured":"Bickel, P.J., Sarkar, P.: Hypothesis testing for automated community detection in networks. J. R. Stat. Soc. Ser. B 78(1), 253\u2013273 (2016)","journal-title":"J. R. Stat. Soc. Ser. B"},{"issue":"7","key":"9946_CR6","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1109\/34.865189","volume":"22","author":"C Biernacki","year":"2000","unstructured":"Biernacki, C., Celeux, G., Govaert, G.: Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intell. 22(7), 719\u2013725 (2000)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"9946_CR7","doi-asserted-by":"crossref","first-page":"1616","DOI":"10.1109\/TKDE.2018.2807452","volume":"30","author":"H Cai","year":"2018","unstructured":"Cai, H., Zheng, V.W., Chang, K.C.: A comprehensive survey of graph embedding: problems, techniques, and applications. IEEE Trans. Knowl. Data Eng. 30, 1616\u20131637 (2018)","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"9946_CR8","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1214\/12-EJS729","volume":"6","author":"A Celisse","year":"2012","unstructured":"Celisse, A., Daudin, J., Pierre, L.: Consistency of maximum-likelihood and variational estimators in the stochastic block model. Electron. J. Stat. 6, 1847\u20131899 (2012)","journal-title":"Electron. J. Stat."},{"issue":"1","key":"9946_CR9","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1214\/14-AOS1272","volume":"43","author":"S Chatterjee","year":"2015","unstructured":"Chatterjee, S.: Matrix estimation by universal singular value thresholding. Ann. Stat. 43(1), 177\u2013214 (2015)","journal-title":"Ann. Stat."},{"issue":"521","key":"9946_CR10","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1080\/01621459.2016.1246365","volume":"113","author":"K Chen","year":"2018","unstructured":"Chen, K., Lei, J.: Network cross-validation for determining the number of communities in network data. J. Am. Stat. Assoc. 113(521), 241\u2013251 (2018)","journal-title":"J. Am. Stat. Assoc."},{"issue":"6","key":"9946_CR11","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1177\/1471082X15577017","volume":"15","author":"E C\u00f4me","year":"2015","unstructured":"C\u00f4me, E., Latouche, P.: Model selection and clustering in stochastic block models based on the exact integrated complete data likelihood. Stat. Model. 15(6), 564\u2013589 (2015)","journal-title":"Stat. Model."},{"key":"9946_CR12","unstructured":"Dahl, D.B.: An improved merge-split sampler for conjugate Dirichlet process mixture models. Tech. Rep. 1086, Department of Statistics, University of Wisconsin, Madison (2003)"},{"issue":"1","key":"9946_CR13","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s11222-006-5338-6","volume":"16","author":"P Dellaportas","year":"2006","unstructured":"Dellaportas, P., Papageorgiou, I.: Multivariate mixtures of normals with unknown number of components. Stat. Comput. 16(1), 57\u201368 (2006)","journal-title":"Stat. Comput."},{"key":"9946_CR14","doi-asserted-by":"crossref","unstructured":"Dhillon, I.S.: Co-clustering documents and words using bipartite spectral graph partitioning. In: Proceedings of the seventh ACM SIGKDD international conference on knowledge discovery and data mining. pp. 269\u2013274. KDD \u201901, ACM, New York, NY, USA (2001)","DOI":"10.1145\/502512.502550"},{"issue":"2","key":"9946_CR15","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/BF01897164","volume":"5","author":"EB Fowlkes","year":"1988","unstructured":"Fowlkes, E.B., Gnanadesikan, R., Kettenring, J.R.: Variable selection in clustering. J. Classif. 5(2), 205\u2013228 (1988)","journal-title":"J. Classif."},{"issue":"1","key":"9946_CR16","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1080\/10618600.2015.1096790","volume":"26","author":"D Franco Salda\u00f1a","year":"2017","unstructured":"Franco Salda\u00f1a, D., Yu, Y., Feng, Y.: How many communities are there? J. Comput. Gr. Stat. 26(1), 171\u2013181 (2017)","journal-title":"J. Comput. Gr. Stat."},{"issue":"2","key":"9946_CR17","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1214\/09-BA414","volume":"4","author":"A Fritsch","year":"2009","unstructured":"Fritsch, A., Ickstadt, K.: Improved criteria for clustering based on the posterior similarity matrix. Bayesian Anal. 4(2), 367\u2013391 (2009)","journal-title":"Bayesian Anal."},{"issue":"3","key":"9946_CR18","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1080\/03610920903140197","volume":"39","author":"G Govaert","year":"2010","unstructured":"Govaert, G., Nadif, M.: Latent block model for contingency table. Commun. Stat. Theo. Methods 39(3), 416\u2013425 (2010)","journal-title":"Commun. Stat. Theo. Methods"},{"issue":"4","key":"9946_CR19","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1093\/biomet\/82.4.711","volume":"82","author":"PJ Green","year":"1995","unstructured":"Green, P.J.: Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711\u2013732 (1995)","journal-title":"Biometrika"},{"issue":"460","key":"9946_CR20","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1198\/016214502388618906","volume":"97","author":"PD Hoff","year":"2002","unstructured":"Hoff, P.D., Raftery, A.E., Handcock, M.S.: Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97(460), 1090\u20131098 (2002)","journal-title":"J. Am. Stat. Assoc."},{"issue":"2","key":"9946_CR21","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","volume":"5","author":"PW Holland","year":"1983","unstructured":"Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109\u2013137 (1983)","journal-title":"Soc. Netw."},{"issue":"1","key":"9946_CR22","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01908075","volume":"2","author":"L Hubert","year":"1985","unstructured":"Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193\u2013218 (1985)","journal-title":"J. Classif."},{"issue":"1","key":"9946_CR23","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1198\/1061860043001","volume":"13","author":"S Jain","year":"2004","unstructured":"Jain, S., Neal, R.M.: A split-merge Markov Chain Monte Carlo procedure for the Dirichlet process mixture model. J. Comput. Gr. Stat. 13(1), 158\u2013182 (2004)","journal-title":"J. Comput. Gr. Stat."},{"issue":"1","key":"9946_CR24","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1214\/088342305000000016","volume":"20","author":"A Jasra","year":"2005","unstructured":"Jasra, A., Holmes, C.C., Stephens, D.A.: Markov Chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Stat. Sci. 20(1), 50\u201367 (2005)","journal-title":"Stat. Sci."},{"key":"9946_CR25","series-title":"Principal component analysis","volume-title":"Springer series in statistics","author":"IT Jolliffe","year":"2002","unstructured":"Jolliffe, I.T.: Springer series in statistics. Principal component analysis. Springer, Berlin (2002)"},{"issue":"9","key":"9946_CR26","doi-asserted-by":"crossref","first-page":"1154","DOI":"10.1109\/TPAMI.2004.71","volume":"26","author":"MHC Law","year":"2004","unstructured":"Law, M.H.C., Figueiredo, M.A.T., Jain, A.K.: Simultaneous feature selection and clustering using mixture models. IEEE Trans. Pattern Anal. Mach. Intell. 26(9), 1154\u20131166 (2004)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"1","key":"9946_CR27","first-page":"401","volume":"44","author":"J Lei","year":"2016","unstructured":"Lei, J.: A goodness-of-fit test for stochastic block models. Ann. Stat. 44(1), 401\u2013424 (2016)","journal-title":"Ann. Stat."},{"issue":"1","key":"9946_CR28","first-page":"215","volume":"43","author":"J Lei","year":"2015","unstructured":"Lei, J., Rinaldo, A.: Consistency of spectral clustering in stochastic block models. Ann. Stat. 43(1), 215\u2013237 (2015)","journal-title":"Ann. Stat."},{"issue":"427","key":"9946_CR29","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1080\/01621459.1994.10476829","volume":"89","author":"JS Liu","year":"1994","unstructured":"Liu, J.S.: The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem. J. Am. Stat. Assoc. 89(427), 958\u2013966 (1994)","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"9946_CR30","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.physrep.2013.08.002","volume":"533","author":"FD Malliaros","year":"2013","unstructured":"Malliaros, F.D., Vazirgiannis, M.: Clustering and community detection in directed networks: a survey. Phys. Rep. 533(4), 95\u2013142 (2013)","journal-title":"Phys. Rep."},{"issue":"3","key":"9946_CR31","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1111\/j.1541-0420.2008.01160.x","volume":"65","author":"C Maugis","year":"2009","unstructured":"Maugis, C., Celeux, G., Martin-Magniette, M.L.: Variable selection for clustering with Gaussian mixture models. Biometrics 65(3), 701\u2013709 (2009)","journal-title":"Biometrics"},{"issue":"8","key":"9946_CR32","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1093\/bioinformatics\/bth068","volume":"20","author":"M Medvedovic","year":"2004","unstructured":"Medvedovic, M., Yeung, K.Y., Bumgarner, R.E.: Bayesian mixture model based clustering of replicated microarray data. Bioinformatics 20(8), 1222\u20131232 (2004)","journal-title":"Bioinformatics"},{"key":"9946_CR33","volume-title":"Bayesian Statistics","author":"K Mengersen","year":"1996","unstructured":"Mengersen, K., Robert, C.: Testing for mixtures: a Bayesian entropic approach (with discussion). In: Berger, J., Bernardo, J., Dawid, A., Lindley, D., Smith, A. (eds.) Bayesian Statistics. Oxford University Press, Oxford (1996)"},{"issue":"521","key":"9946_CR34","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1080\/01621459.2016.1255636","volume":"113","author":"JW Miller","year":"2018","unstructured":"Miller, J.W., Harrison, M.T.: Mixture models with a prior on the number of components. J. Am. Stat. Assoc. 113(521), 340\u2013356 (2018)","journal-title":"J. Am. Stat. Assoc."},{"key":"9946_CR35","unstructured":"Murphy, K.P.: Conjugate Bayesian analysis of the Gaussian distribution. Tech. rep. (2007)"},{"issue":"7","key":"9946_CR36","doi-asserted-by":"crossref","first-page":"078301","DOI":"10.1103\/PhysRevLett.117.078301","volume":"117","author":"MEJ Newman","year":"2016","unstructured":"Newman, M.E.J., Reinert, G.: Estimating the number of communities in a network. Phys. Rev. Lett. 117(7), 078301 (2016)","journal-title":"Phys. Rev. Lett."},{"issue":"5","key":"9946_CR37","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1214\/009053604000000788","volume":"32","author":"A Nobile","year":"2004","unstructured":"Nobile, A.: On the posterior distribution of the number of components in a finite mixture. Ann. Stat. 32(5), 2044\u20132073 (2004)","journal-title":"Ann. Stat."},{"issue":"455","key":"9946_CR38","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1198\/016214501753208735","volume":"96","author":"K Nowicki","year":"2001","unstructured":"Nowicki, K., Snijders, T.A.B.: Estimation and prediction for stochastic blockstructures. J. Am. Stat. Assoc. 96(455), 1077\u20131087 (2001)","journal-title":"J. Am. Stat. Assoc."},{"issue":"3","key":"9946_CR39","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s10588-005-5378-z","volume":"11","author":"CE Priebe","year":"2005","unstructured":"Priebe, C.E., Conroy, J.M., Marchette, D.J., Park, Y.: Scan statistics on Enron graphs. Comput. Math. Organ. Theo. 11(3), 229\u2013247 (2005)","journal-title":"Comput. Math. Organ. Theo."},{"issue":"13","key":"9946_CR40","doi-asserted-by":"crossref","first-page":"5995","DOI":"10.1073\/pnas.1814462116","volume":"116","author":"CE Priebe","year":"2019","unstructured":"Priebe, C.E., Park, Y., Vogelstein, J.T., Conroy, J.M., Lyzinski, V., Tang, M., Athreya, A., Cape, J., Bridgeford, E.: On a two-truths phenomenon in spectral graph clustering. Proc. Nat. Acad. Sci. 116(13), 5995\u20136000 (2019)","journal-title":"Proc. Nat. Acad. Sci."},{"issue":"473","key":"9946_CR41","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1198\/016214506000000113","volume":"101","author":"AE Raftery","year":"2006","unstructured":"Raftery, A.E., Dean, N.: Variable selection for model-based clustering. J. Am. Stat. Assoc. 101(473), 168\u2013178 (2006)","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"9946_CR42","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1111\/1467-9868.00095","volume":"59","author":"S Richardson","year":"1997","unstructured":"Richardson, S., Green, P.J.: On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. Roy. Stat. Soc. B 59(4), 731\u2013792 (1997)","journal-title":"J. Roy. Stat. Soc. B"},{"issue":"4","key":"9946_CR43","doi-asserted-by":"crossref","first-page":"1878","DOI":"10.1214\/11-AOS887","volume":"39","author":"K Rohe","year":"2011","unstructured":"Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878\u20131915 (2011)","journal-title":"Ann. Stat."},{"key":"9946_CR44","doi-asserted-by":"crossref","unstructured":"Rohe, K., Qin, T., Yu, B.: Co-clustering directed graphs to discover asymmetries and directional communities. In: Proceedings of the National Academy of Sciences (2016)","DOI":"10.1073\/pnas.1525793113"},{"key":"9946_CR45","doi-asserted-by":"crossref","unstructured":"Rubin-Delanchy, P., Adams, N.M., Heard, N.A.: Disassortativity of computer networks. In: 2016 IEEE conference on intelligence and security informatics (ISI). pp. 243\u2013247, (2016)","DOI":"10.1109\/ISI.2016.7745482"},{"key":"9946_CR46","unstructured":"Rubin-Delanchy, P., Priebe, C.E., Tang, M., Cape, J.: A statistical interpretation of spectral embedding: the generalised random dot product graph. ArXiv e-prints (2017)"},{"issue":"1","key":"9946_CR47","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s003579900004","volume":"14","author":"TAB Snijders","year":"1997","unstructured":"Snijders, T.A.B., Nowicki, K.: Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classif. 14(1), 75\u2013100 (1997)","journal-title":"J. Classif."},{"issue":"1","key":"9946_CR48","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1214\/aos\/1016120364","volume":"28","author":"M Stephens","year":"2000","unstructured":"Stephens, M.: Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods. Ann. Stat. 28(1), 40\u201374 (2000)","journal-title":"Ann. Stat."},{"issue":"1","key":"9946_CR49","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1109\/TPAMI.2013.135","volume":"36","author":"DL Sussman","year":"2014","unstructured":"Sussman, D.L., Tang, M., Priebe, C.E.: Consistent latent position estimation and vertex classification for random dot product graphs. IEEE Trans. Pattern Anal. Mach. Intell. 36(1), 48\u201357 (2014)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"470","key":"9946_CR50","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1198\/016214504000001565","volume":"100","author":"MG Tadesse","year":"2005","unstructured":"Tadesse, M.G., Sha, N., Vannucci, M.: Bayesian variable selection in clustering high-dimensional data. J. Am. Stat. Assoc. 100(470), 602\u2013617 (2005)","journal-title":"J. Am. Stat. Assoc."},{"issue":"5","key":"9946_CR51","doi-asserted-by":"crossref","first-page":"2360","DOI":"10.1214\/17-AOS1623","volume":"46","author":"M Tang","year":"2018","unstructured":"Tang, M., Priebe, C.E.: Limit theorems for eigenvectors of the normalized Laplacian for random graphs. Ann. Stat. 46(5), 2360\u20132415 (2018)","journal-title":"Ann. Stat."},{"issue":"3","key":"9946_CR52","doi-asserted-by":"crossref","first-page":"1406","DOI":"10.1214\/13-AOS1112","volume":"41","author":"M Tang","year":"2013","unstructured":"Tang, M., Sussman, D.L., Priebe, C.E.: Universally consistent vertex classification for latent positions graphs. Ann. Stat. 41(3), 1406\u20131430 (2013)","journal-title":"Ann. Stat."},{"issue":"4","key":"9946_CR53","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11222-007-9033-z","volume":"1","author":"U von Luxburg","year":"2007","unstructured":"von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 1(4), 395\u2013416 (2007)","journal-title":"Stat. Comput."},{"issue":"397","key":"9946_CR54","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1080\/01621459.1987.10478385","volume":"82","author":"WJ Wang","year":"1987","unstructured":"Wang, W.J., Wong, G.Y.: Stochastic blockmodels for directed graphs. J. Am. Stat. Assoc. 82(397), 8\u201319 (1987)","journal-title":"J. Am. Stat. Assoc."},{"issue":"490","key":"9946_CR55","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1198\/jasa.2010.tm09415","volume":"105","author":"DM Witten","year":"2010","unstructured":"Witten, D.M., Tibshirani, R.: A framework for feature selection in clustering. J. Am. Stat. Assoc. 105(490), 713\u2013726 (2010)","journal-title":"J. Am. Stat. Assoc."},{"key":"9946_CR56","unstructured":"Wolfe, P.J., Olhede, S.C.: Nonparametric graphon estimation. arXiv e-prints (2013)"},{"issue":"2","key":"9946_CR57","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s11222-011-9233-4","volume":"22","author":"J Wyse","year":"2012","unstructured":"Wyse, J., Friel, N.: Block clustering with collapsed latent block models. Stat. Comput. 22(2), 415\u2013428 (2012)","journal-title":"Stat. Comput."},{"issue":"1","key":"9946_CR58","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1017\/nws.2016.25","volume":"5","author":"J Wyse","year":"2017","unstructured":"Wyse, J., Friel, N., Latouche, P.: Inferring structure in bipartite networks using the latent blockmodel and exact ICL. Net. Sci. 5(1), 45\u201369 (2017)","journal-title":"Net. Sci."},{"key":"9946_CR59","doi-asserted-by":"crossref","unstructured":"Yang, C., Priebe, C.E., Park, Y., Marchette, D.J.: Simultaneous dimensionality and complexity model selection for spectral graph clustering. arXiv e-prints arXiv:1904.02926 (2019)","DOI":"10.1080\/10618600.2020.1824870"},{"key":"9946_CR60","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1007\/978-3-540-77004-6_11","volume-title":"Algorithms Models Web Graph","author":"SJ Young","year":"2007","unstructured":"Young, S.J., Scheinerman, E.R.: Random dot product graph models for social networks. In: Bonato, A., Chung, F.R.K. (eds.) Algorithms Models Web Graph, pp. 138\u2013149. Springer, Berlin (2007)"},{"issue":"18","key":"9946_CR61","doi-asserted-by":"crossref","first-page":"7321","DOI":"10.1073\/pnas.1006642108","volume":"108","author":"Y Zhao","year":"2011","unstructured":"Zhao, Y., Levina, E., Zhu, J.: Community extraction for social networks. Proc. Nat. Acad. Sci. 108(18), 7321\u20137326 (2011)","journal-title":"Proc. Nat. Acad. Sci."},{"key":"9946_CR62","doi-asserted-by":"crossref","unstructured":"Zheng, Q., Skillicorn, D.B.: Spectral embedding of directed networks. 2015 IEEE\/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM) pp. 432\u2013439 (2015)","DOI":"10.1145\/2808797.2809310"},{"issue":"2","key":"9946_CR63","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.csda.2005.09.010","volume":"51","author":"M Zhu","year":"2006","unstructured":"Zhu, M., Ghodsi, A.: Automatic dimensionality selection from the scree plot via the use of profile likelihood. Comput. Stat. Data Anal. 51(2), 918\u2013930 (2006)","journal-title":"Comput. Stat. Data Anal."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-020-09946-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-020-09946-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-020-09946-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T23:19:57Z","timestamp":1622071197000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-020-09946-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,27]]},"references-count":63,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["9946"],"URL":"https:\/\/doi.org\/10.1007\/s11222-020-09946-6","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,5,27]]},"assertion":[{"value":"12 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}