{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T09:38:00Z","timestamp":1790674680236,"version":"4.1.0"},"reference-count":68,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2022,6,2]],"date-time":"2022-06-02T00:00:00Z","timestamp":1654128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,2,11]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>We study the properties of two subsampling procedures for networks, vertex subsampling and $p$-subsampling, under the sparse graphon model. The consistency of network subsampling is demonstrated under the minimal assumptions of weak convergence of the corresponding network statistics and an expected subsample size growing to infinity more slowly than the number of vertices in the network. Furthermore, under appropriate sparsity conditions, we derive limiting distributions for the nonzero eigenvalues of an adjacency matrix under the sparse graphon model. Our weak convergence result implies the consistency of our subsampling procedures for eigenvalues under appropriate conditions.<\/jats:p>","DOI":"10.1093\/biomet\/asac032","type":"journal-article","created":{"date-parts":[[2022,6,2]],"date-time":"2022-06-02T09:47:27Z","timestamp":1654163247000},"page":"15-32","source":"Crossref","is-referenced-by-count":12,"title":["Subsampling sparse graphons under minimal assumptions"],"prefix":"10.1093","volume":"110","author":[{"given":"Robert","family":"Lunde","sequence":"first","affiliation":[{"name":"University of Michigan Department of Statistics, , 1085 South University, Ann Arbor, Michigan 48109, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Purnamrita","family":"Sarkar","sequence":"additional","affiliation":[{"name":"University of Texas Department of Statistics and Data Sciences, , 105 E. 24th Street, Austin, Texas 78712, U.S.A"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,6,2]]},"reference":[{"key":"2023060200102069800_B1","first-page":"1981","article-title":"Mixed membership stochastic blockmodels","volume":"9","author":"Airoldi,","year":"2008","journal-title":"J. Mach. Learn. Res."},{"key":"2023060200102069800_B2","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1016\/0047-259X(81)90099-3","article-title":"Representations for partially exchangeable arrays of random variables","volume":"11","author":"Aldous,","year":"1981","journal-title":"J. Mult. Anal."},{"key":"2023060200102069800_B3","doi-asserted-by":"crossref","DOI":"10.1038\/srep28955","article-title":"Comparison of large networks with sub-sampling strategies","volume":"6","author":"Ali,","year":"2016","journal-title":"Sci. Rep."},{"key":"2023060200102069800_B4","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1214\/17-AOS1545","article-title":"On semidefinite relaxations for the block model","volume":"46","author":"Amini,","year":"2018","journal-title":"Ann. Statist."},{"key":"2023060200102069800_B5","article-title":"A simple sublinear-time algorithm for counting arbitrary subgraphs via edge sampling","volume-title":"Proc. 10th Conf. Innovations in Theoretical Computer Science (ITCS 2019)","author":"Assadi,","year":"2019"},{"key":"2023060200102069800_B6","first-page":"623","article-title":"Reductions in streaming algorithms, with an application to counting triangles in graphs","volume-title":"Proc. 13th ACM-SIAM Symp. Discrete Algorithms (SODA \u201902)","author":"Bar-Yosseff,","year":"2002"},{"key":"2023060200102069800_B7","first-page":"1","article-title":"Towards tighter space bounds for counting triangles and other substructures in graph streams","volume-title":"34th Symp. Theoretical Aspects of Computer Science (STACS 2017)","author":"Bera,","year":"2017"},{"key":"2023060200102069800_B8","doi-asserted-by":"crossref","first-page":"2384","DOI":"10.1214\/15-AOS1338","article-title":"Subsampling bootstrap of count features of networks","volume":"43","author":"Bhattacharyya,","year":"2015","journal-title":"Ann. Statist."},{"key":"2023060200102069800_B9","doi-asserted-by":"crossref","first-page":"21068","DOI":"10.1073\/pnas.0907096106","article-title":"A nonparametric view of network models and Newman-Girvan and other modularities","volume":"106","author":"Bickel,","year":"2009","journal-title":"Proc. Nat. Acad. Sci."},{"key":"2023060200102069800_B10","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1214\/11-AOS904","article-title":"The method of moments and degree distributions for network models","volume":"39","author":"Bickel,","year":"2011","journal-title":"Ann. Statist."},{"key":"2023060200102069800_B11","first-page":"7740","article-title":"Sparse exchangeable graphs and their limits via graphon processes","volume":"18","author":"Borgs,","year":"2017","journal-title":"J. Mach. Learn. Res."},{"key":"2023060200102069800_B12","doi-asserted-by":"crossref","first-page":"3019","DOI":"10.1090\/tran\/7543","article-title":"An $L^p$ theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions","volume":"372","author":"Borgs,","year":"2019","journal-title":"Trans. Am. Math. Soc."},{"key":"2023060200102069800_B13","doi-asserted-by":"crossref","first-page":"151","DOI":"10.4007\/annals.2012.176.1.2","article-title":"Convergent sequences of dense graphs II: Multiway cuts and statistical physics","volume":"176","author":"Borgs,","year":"2012","journal-title":"Ann. Math."},{"key":"2023060200102069800_B14","doi-asserted-by":"crossref","first-page":"1311","DOI":"10.1080\/01621459.2017.1341413","article-title":"Edge exchangeable models for interaction networks","volume":"113","author":"Crane,","year":"2018","journal-title":"J. Am. Statist. Assoc."},{"key":"2023060200102069800_B15","volume-title":"Introduction to Hilbert Spaces with Applications","author":"Debnath,","year":"2005"},{"key":"2023060200102069800_B16","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1002\/rsa.20965","article-title":"Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon","volume":"58","author":"Delmas,","year":"2021","journal-title":"Random Struct. Algor."},{"key":"2023060200102069800_B17","doi-asserted-by":"crossref","first-page":"1603","DOI":"10.1137\/15M1054389","article-title":"Approximately counting triangles in sublinear time","volume":"46","author":"Eden,","year":"2017","journal-title":"SIAM J. Comp,"},{"key":"2023060200102069800_B18","first-page":"321","article-title":"Unperturbed: Spectral analysis beyond Davis-Kahan","volume":"83","author":"Eldridge,","year":"2018","journal-title":"Proc. Mach. Learn. Res."},{"key":"2023060200102069800_B19","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/S0097539704447304","article-title":"On sums of independent random variables with unbounded variance, and estimating the average degree in a graph","volume":"35","author":"Feige,","year":"2006","journal-title":"SIAM J. Comp."},{"key":"2023060200102069800_B20","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/BF02579329","article-title":"The eigenvalues of random symmetric matrices","volume":"1","author":"F\u00fcredi,","year":"1981","journal-title":"Combinatorica"},{"key":"2023060200102069800_B21","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1214\/19-STS736","article-title":"Minimax rates in network analysis: Graphon estimation, community detection and hypothesis testing","volume":"36","author":"Gao,","year":"2021","journal-title":"Statist. Sci."},{"key":"2023060200102069800_B22","doi-asserted-by":"crossref","DOI":"10.1007\/s41109-017-0042-3","article-title":"Identifying network structure similarity using spectral graph theory","volume":"3","author":"Gera,","year":"2018","journal-title":"Appl. Network Sci."},{"key":"2023060200102069800_B23","article-title":"Two-sample tests for large random graphs using network statistics","author":"Ghoshdastidar,","year":"2017","journal-title":"Conf. Learning Theory (COLT)"},{"key":"2023060200102069800_B24","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1002\/rsa.20203","article-title":"Approximating average parameters of graphs","volume":"32","author":"Goldreich,","year":"2008","journal-title":"Random Struct. Algor."},{"key":"2023060200102069800_B25","first-page":"99","article-title":"Counting stars and other small subgraphs in sublinear time","volume-title":"Proc. 21st ACM-SIAM Symp. Discrete Algorithms (SODA)","author":"Gonen,","year":"2010"},{"key":"2023060200102069800_B26","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1214\/21-EJS1896","article-title":"Bootstrapping exchangeable random graphs","volume":"16","author":"Green,","year":"2022","journal-title":"Electron. J. Statist."},{"key":"2023060200102069800_B27","first-page":"723","article-title":"A kernel two-sample test","volume":"13","author":"Gretton,","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"2023060200102069800_B28","doi-asserted-by":"crossref","first-page":"1025","DOI":"10.1007\/s00440-015-0659-z","article-title":"Community detection in sparse networks via Grothendieck\u2019s inequality","volume":"165","author":"Gu\u00e9don,","year":"2016","journal-title":"Prob. Theory Rel. Fields"},{"key":"2023060200102069800_B29","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1198\/016214502388618906","article-title":"Latent space approaches to social network analysis","volume":"97","author":"Hoff,","year":"2002","journal-title":"J. Am. Statist. Assoc."},{"key":"2023060200102069800_B30","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":"2023060200102069800_B31","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1090\/S0273-0979-06-01126-8","article-title":"Expander graphs and their applications","volume":"43","author":"Hoory,","year":"2006","journal-title":"Bull. Am. Math. Soc."},{"key":"2023060200102069800_B32","article-title":"Relations on probability spaces arrays of random variables","author":"Hoover,","year":"1979"},{"key":"2023060200102069800_B33","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevResearch.2.043352","article-title":"Emergence of complex socioeconomic networks driven by individual and collective interests","volume":"2","author":"Iranzo,","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"2023060200102069800_B34","first-page":"1778","article-title":"A hybrid sampling scheme for triangle counting","volume-title":"Proc. 28th ACM-SIAM Symp. Discrete Algorithms (SODA 2017, Barcelona, Spain)","author":"Kallaugher,","year":"2017"},{"key":"2023060200102069800_B35","first-page":"598","article-title":"Counting arbitrary subgraphs in data streams","volume-title":"Proc. 39th Int. Colloq. Automata, Languages, and Programming (ICALP 2012, Paris, France)","author":"Kane,","year":"2012"},{"key":"2023060200102069800_B36","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.83.016107","article-title":"Stochastic blockmodels and community structure in networks","volume":"83","author":"Karrer,","year":"2011","journal-title":"Phys. Rev. E"},{"key":"2023060200102069800_B37","first-page":"1","article-title":"Random matrix approximation of spectra of integral operators","volume":"6","author":"Koltchinskii,","year":"2001","journal-title":"Bernoulli"},{"key":"2023060200102069800_B38","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1214\/20-AOS1976","article-title":"Network representation using graph root distributions","volume":"49","author":"Lei,","year":"2021","journal-title":"Ann. Statist."},{"key":"2023060200102069800_B39","article-title":"Bootstrapping networks with latent space structure","author":"Levin,","year":"2021"},{"key":"2023060200102069800_B40","doi-asserted-by":"publisher","DOI":"10.3390\/brainsci10020092","article-title":"The maximum eigenvalue of the brain functional network adjacency matrix: Meaning and application in mental fatigue evaluation","volume":"10","author":"Li,","year":"2020","journal-title":"Brain Sci."},{"key":"2023060200102069800_B41","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/060","volume-title":"Large Networks and Graph Limits","author":"Lov\u00e1sz,","year":"2012"},{"key":"2023060200102069800_B42","first-page":"311","article-title":"Expansion properties of large social graphs","volume-title":"Int. Conf. Database Systems for Advanced Applications (DASFAA 2011)","author":"Malliaros,","year":"2011"},{"key":"2023060200102069800_B43","article-title":"Central limit theorems for local network statistics","author":"Maugis,","year":"2020"},{"key":"2023060200102069800_B44","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1145\/2902251.2902283","article-title":"Better algorithms for counting triangles in data streams","volume-title":"Proc. 35th ACM SIGMOD-SIGACT-SIGAI Symp. Principles of Database Systems (PODS \u201916, San Francisco, California)","author":"McGregor,","year":"2016"},{"key":"2023060200102069800_B45","first-page":"119","article-title":"The complexity of counting cycles in the adjacency list streaming model","volume-title":"Proc. 38th ACM SIGMOD-SIGACT-SIGAI Symp. Principles of Database Systems (PODS \u201919, Amsterdam, Netherlands)","author":"McGregor,","year":"2019"},{"key":"2023060200102069800_B46","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511921681","volume-title":"Graph Spectra for Complex Networks","author":"Mieghem,","year":"2010"},{"key":"2023060200102069800_B47","first-page":"7071","article-title":"On clustering network-valued data","volume-title":"Proc. 31st Int. Conf. Neural Information Processing Systems (NIPS\u201917, Long Beach, California)","author":"Mukherjee,","year":"2017"},{"key":"2023060200102069800_B48","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1214\/20-EJS1789","article-title":"Bootstrap estimators for the tail-index and for the count statistics of graphex processes","volume":"15","author":"Naulet,","year":"2021","journal-title":"Electron. J. Statist."},{"key":"2023060200102069800_B49","article-title":"Subsampling large graphs and invariance in networks","author":"Orbanz,","year":"2017"},{"key":"2023060200102069800_B50","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.laa.2017.11.014","article-title":"Random perturbation of low rank matrices: Improving classical bounds","volume":"540","author":"O\u2019Rourke,","year":"2018","journal-title":"Lin. Algeb. Applic."},{"key":"2023060200102069800_B51","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1109\/SAMPTA.2017.8024399","article-title":"A semidefinite program for unbalanced multisection in the stochastic block model","volume-title":"2017 Int. Conf. Sampling Theory and Applications (SampTA)","author":"Perry,","year":"2017"},{"key":"2023060200102069800_B52","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1554-7","volume-title":"Subsampling","author":"Politis,","year":"1999"},{"key":"2023060200102069800_B53","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1017\/nws.2016.23","article-title":"Properties of latent variable network models","volume":"4","author":"Rastelli,","year":"2016","journal-title":"Network Sci."},{"key":"2023060200102069800_B54","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.71.036151","article-title":"Onset of synchronization in large networks of coupled oscillators","volume":"71","author":"Restrepo,","year":"2005","journal-title":"Phys. Rev. E"},{"key":"2023060200102069800_B55","doi-asserted-by":"crossref","DOI":"10.1111\/rssb.12509","article-title":"A statistical interpretation of spectral embedding: The generalised random dot product graph","author":"Rubin-Delanchy,","year":"2022","journal-title":"J. R. Statist. Soc. B"},{"key":"2023060200102069800_B56","article-title":"Lecture 3. Yale Computer Science, CPSC 662: Spectral Graph Theory","author":"Spielman,","year":"2012"},{"key":"2023060200102069800_B57","article-title":"The eigenvalues of stochastic blockmodel graphs","author":"Tang,","year":"2018"},{"key":"2023060200102069800_B58","doi-asserted-by":"crossref","first-page":"1599","DOI":"10.3150\/15-BEJ789","article-title":"A nonparametric two-sample hypothesis testing problem for random graphs","volume":"23","author":"Tang,","year":"2017","journal-title":"Bernoulli"},{"key":"2023060200102069800_B59","doi-asserted-by":"crossref","first-page":"4165","DOI":"10.1016\/j.physa.2011.12.021","article-title":"Social structure of Facebook networks","volume":"391","author":"Traud,","year":"2012","journal-title":"Physica A"},{"key":"2023060200102069800_B60","first-page":"837","article-title":"DOULION: Counting triangles in massive graphs with a coin","volume-title":"Proc. 15th ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining (KDD \u201909, Paris, France)","author":"Tsourakakis,","year":"2009"},{"key":"2023060200102069800_B61","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TNET.2008.925623","article-title":"Virus spread in networks","volume":"17","author":"Van Mieghem,","year":"2009","journal-title":"IEEE\/ACM Trans. Networking"},{"key":"2023060200102069800_B62","article-title":"The class of random graphs arising from exchangeable random measures","author":"Veitch,","year":"2015"},{"key":"2023060200102069800_B63","doi-asserted-by":"crossref","first-page":"3274","DOI":"10.1214\/18-AOS1778","article-title":"Sampling and estimation for (sparse) exchangeable random graphs","volume":"47","author":"Veitch,","year":"2019","journal-title":"Ann. Statist."},{"key":"2023060200102069800_B64","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s00493-007-2190-z","article-title":"Spectral norm of random matrices","volume":"27","author":"Vu,","year":"2007","journal-title":"Combinatorica"},{"key":"2023060200102069800_B65","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":"Ann. Statist."},{"key":"2023060200102069800_B66","first-page":"1185","article-title":"Provable estimation of the number of blocks in block models","volume-title":"Proc. 21st Int. Conf. Artificial Intelligence and Statistics","author":"Yan,","year":"2018"},{"key":"2023060200102069800_B67","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1007\/978-3-540-77004-6_11","article-title":"Random dot product graph models for social networks","volume-title":"Proc. 5th Int. Conf. Algorithms and Models for the Web-Graph (WAW\u201907)","author":"Young,","year":"2007"},{"key":"2023060200102069800_B68","first-page":"726","article-title":"Edgeworth expansions for network moments","volume":"2","author":"Zhang,","year":"2022","journal-title":"Ann. Statist."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asac032\/45402711\/asac032.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/1\/15\/49160060\/asac032.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/110\/1\/15\/49160060\/asac032.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T16:24:36Z","timestamp":1685723076000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/110\/1\/15\/6598813"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,2]]},"references-count":68,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,6,2]]},"published-print":{"date-parts":[[2023,2,11]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asac032","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,3,1]]},"published":{"date-parts":[[2022,6,2]]}}}