{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:50:27Z","timestamp":1791323427022,"version":"4.3.1"},"reference-count":27,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"Large-scale network inference with uncertainty quantification has important applications in natural, social and medical sciences. The recent work of Fan et al. (J. R. Stat. Soc. Ser. B. Stat. Methodol. 84 (2022) 630\u2013653) introduced a general framework of statistical inference on membership profiles in large networks (SIMPLE) for testing the sharp null hypothesis that a pair of given nodes share the same membership profile. In real applications, there are often groups of nodes under investigation that may share similar membership profiles in the presence of relatively weaker signals than the setting considered in SIMPLE. To address these practical challenges, in this paper, we propose a SIMPLE method with random coupling (SIMPLE-RC) for testing the nonsharp composite null hypothesis that a group of given nodes shares similar (not necessarily identical) membership profiles under weaker signals. Utilizing the idea of random coupling, we construct our test as the maximum of the SIMPLE tests for subsampled node pairs from the group. Such a technique significantly reduces the correlation among individual SIMPLE tests while largely maintaining the power, enabling delicate analysis of the asymptotic distributions of the SIMPLE-RC test. Our method and theory cover both the cases with and without node degree heterogeneity. These new theoretical developments are empowered by a second-order expansion of spiked eigenvectors under the \u2113\u221e-norm, built upon our work for random matrices with weak spikes. Our theoretical results and the practical advantages of the newly proposed method are demonstrated through simulations and real data examples.<\/jats:p>","DOI":"10.1214\/26-aos2655","type":"journal-article","created":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T12:53:06Z","timestamp":1790945586000},"page":"2507-2531","source":"Crossref","is-referenced-by-count":0,"title":["SIMPLE-RC: Group network inference with NonSharp nulls and weak signals"],"prefix":"10.1214","volume":"54","author":[{"given":"Jianqing","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Operations Research & Financial Engineering, Princeton University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yingying","family":"Fan","sequence":"additional","affiliation":[{"name":"Department of Data Sciences and Operations, University of Southern California"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinchi","family":"Lv","sequence":"additional","affiliation":[{"name":"Department of Data Sciences and Operations, University of Southern California"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fan","family":"Yang","sequence":"additional","affiliation":[{"name":"Yau Mathematical Sciences Center, Tsinghua University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"14","doi-asserted-by":"publisher","unstructured":"HE, Y.<\/small>, KNOWLES, A.<\/small> and ROSENTHAL, R.<\/small> (2018). Isotropic self-consistent equations for mean-field random matrices. Probab. Theory Related Fields<\/i> 171<\/b> 203\u2013249.","DOI":"10.1007\/s00440-017-0776-y"},{"key":"9","doi-asserted-by":"publisher","unstructured":"FAN, J.<\/small>, FAN, Y.<\/small>, HAN, X.<\/small> and LV, J.<\/small> (2022). Asymptotic theory of eigenvectors for random matrices with diverging spikes. J. Amer. Statist. Assoc.<\/i> 117<\/b> 996\u20131009.","DOI":"10.1080\/01621459.2020.1840990"},{"key":"8","doi-asserted-by":"publisher","unstructured":"FAN, J.<\/small>, FAN, Y.<\/small>, HAN, X.<\/small> and LV, J.<\/small> (2022). SIMPLE: Statistical inference on membership profiles in large networks. J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> 84<\/b> 630\u2013653.","DOI":"10.1111\/rssb.12505"},{"key":"16","doi-asserted-by":"publisher","unstructured":"JIN, J.<\/small>, KE, Z. T.<\/small> and LUO, S.<\/small> (2021). Optimal adaptivity of signed-polygon statistics for network testing. Ann. Statist.<\/i> 49<\/b> 3408\u20133433.","DOI":"10.1214\/21-aos2089"},{"key":"25","doi-asserted-by":"publisher","unstructured":"MCDAID, A. F.<\/small>, MURPHY, T. B.<\/small>, FRIEL, N.<\/small> and HURLEY, N. J.<\/small> (2013). Improved Bayesian inference for the stochastic block model with application to large networks. Comput. Statist. Data Anal.<\/i> 60<\/b> 12\u201331.","DOI":"10.1016\/j.csda.2012.10.021"},{"key":"24","doi-asserted-by":"publisher","unstructured":"LI, T.<\/small>, LEVINA, E.<\/small> and ZHU, J.<\/small> (2020). Network cross-validation by edge sampling. Biometrika<\/i> 107<\/b> 257\u2013276.","DOI":"10.1093\/biomet\/asaa006"},{"key":"6","doi-asserted-by":"publisher","unstructured":"BLOEMENDAL, A.<\/small>, ERD\u0150S, L.<\/small>, KNOWLES, A.<\/small>, YAU, H.-T.<\/small> and YIN, J.<\/small> (2014). Isotropic local laws for sample covariance and generalized Wigner matrices. Electron. J. Probab.<\/i> 19<\/b> no. 33, 53.","DOI":"10.1214\/ejp.v19-3054"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Jin, J.<\/small> (2015). Fast community detection by SCORE. Ann. Statist.<\/i> 43<\/b> 57\u201389.","DOI":"10.1214\/14-AOS1265"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Rohe, K.<\/small>, Chatterjee, S.<\/small> and Yu, B.<\/small> (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist.<\/i> 39<\/b> 1878\u20131915.","DOI":"10.1214\/11-AOS887"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Ajanki, O. H.<\/small>, Erd\u0151s, L.<\/small> and Kr\u00fcger, T.<\/small> (2017). Universality for general Wigner-type matrices. Probab. Theory Related Fields<\/i> 169<\/b> 667\u2013727.","DOI":"10.1007\/s00440-016-0740-2"},{"key":"2","unstructured":"Airoldi, E. M.<\/small>, Blei, D. M.<\/small>, Fienberg, S. E.<\/small> and Xing, E. P.<\/small> (2008). Mixed membership stochastic blockmodels. J. Mach. Learn. Res.<\/i> 9<\/b> 1981\u20132014."},{"key":"5","doi-asserted-by":"publisher","unstructured":"Bickel, P. J.<\/small>, Chen, A.<\/small> and Levina, E.<\/small> (2011). The method of moments and degree distributions for network models. Ann. Statist.<\/i> 39<\/b> 2280\u20132301.","DOI":"10.1214\/11-AOS904"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Lei, J.<\/small> and Rinaldo, A.<\/small> (2015). Consistency of spectral clustering in stochastic block models. Ann. Statist.<\/i> 43<\/b> 215\u2013237.","DOI":"10.1214\/14-AOS1274"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Knowles, A.<\/small> and Yin, J.<\/small> (2017). Anisotropic local laws for random matrices. Probab. Theory Related Fields<\/i> 169<\/b> 257\u2013352.","DOI":"10.1007\/s00440-016-0730-4"},{"key":"20","doi-asserted-by":"publisher","unstructured":"LE, C. M.<\/small> and LEVINA, E.<\/small> (2022). Estimating the number of communities by spectral methods. Electron. J. Stat.<\/i> 16<\/b> 3315\u20133342.","DOI":"10.1214\/21-ejs1971"},{"key":"7","doi-asserted-by":"publisher","unstructured":"ERD\u0150S, L.<\/small>, KNOWLES, A.<\/small>, YAU, H.-T.<\/small> and YIN, J.<\/small> (2013). Spectral statistics of Erd\u0151s-R\u00e9nyi graphs I: Local semicircle law. Ann. Probab.<\/i> 41<\/b> 2279\u20132375.","DOI":"10.1214\/11-AOP734"},{"key":"18","doi-asserted-by":"publisher","unstructured":"KNOWLES, A.<\/small> and YIN, J.<\/small> (2013). The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure Appl. Math.<\/i> 66<\/b> 1663\u20131749.","DOI":"10.1002\/cpa.21450"},{"key":"1","doi-asserted-by":"crossref","unstructured":"ABBE, E.<\/small> (2018). Community detection and stochastic block models: Recent developments. J. Mach. Learn. Res.<\/i> 18<\/b> Paper No. 177, 86.","DOI":"10.1561\/9781680834772"},{"key":"17","doi-asserted-by":"publisher","unstructured":"JIN, J.<\/small>, KE, Z. T.<\/small> and LUO, S.<\/small> (2024). Mixed membership estimation for social networks. J. Econometrics<\/i> 239<\/b> Paper No. 105369, 17.","DOI":"10.1016\/j.jeconom.2022.12.003"},{"key":"13","doi-asserted-by":"publisher","unstructured":"HAN, X.<\/small>, YANG, Q.<\/small> and FAN, Y.<\/small> (2023). Universal rank inference via residual subsampling with application to large networks. Ann. Statist.<\/i> 51<\/b> 1109\u20131133.","DOI":"10.1214\/23-aos2282"},{"key":"4","unstructured":"BANERJEE, D.<\/small> and MA, Z.<\/small> (2017). Optimal hypothesis testing for stochastic block models with growing degrees. arXiv preprint. Available at arXiv:1705.05305<\/a>."},{"key":"11","unstructured":"GAO, C.<\/small> and LAFFERTY, J.<\/small> (2017). Testing for global network structure using small subgraph statistics. arXiv preprint. Available at arXiv:1710.00862<\/a>."},{"key":"12","doi-asserted-by":"publisher","unstructured":"HAN, X.<\/small>, TONG, X.<\/small> and FAN, Y.<\/small> (2023). Eigen selection in spectral clustering: A theory-guided practice. J. Amer. Statist. Assoc.<\/i> 118<\/b> 109\u2013121.","DOI":"10.1080\/01621459.2021.1917418"},{"key":"10","doi-asserted-by":"crossref","unstructured":"FAN, J.<\/small>, FAN, Y.<\/small>, LV, J.<\/small> and YANG, F.<\/small> (2026). Supplement to \u201cSIMPLE-RC: Group network inference with nonsharp nulls and weak signals.\u201d https:\/\/doi.org\/10.1214\/26-AOS2655SUPP<\/a>","DOI":"10.1214\/26-AOS2655SUPP"},{"key":"21","doi-asserted-by":"publisher","unstructured":"LE, C. M.<\/small>, LEVINA, E.<\/small> and VERSHYNIN, R.<\/small> (2016). Optimization via low-rank approximation for community detection in networks. Ann. Statist.<\/i> 44<\/b> 373\u2013400.","DOI":"10.1214\/15-AOS1360"},{"key":"23","unstructured":"LEVIN, K.<\/small>, LODHIA, A.<\/small> and LEVINA, E.<\/small> (2022). Recovering shared structure from multiple networks with unknown edge distributions. J. Mach. Learn. Res.<\/i> 23<\/b> Paper No. 3, 48."},{"key":"27","unstructured":"SHAN, Q.<\/small> and LEVINA, E.<\/small> (2022). Network resampling for estimating uncertainty. arXiv preprint. Available at arXiv:2206.13088<\/a>."}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:05:46Z","timestamp":1791320746000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-54\/issue-5\/SIMPLE-RC--Group-network-inference-with-NonSharp-nulls-and\/10.1214\/26-AOS2655.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,1]]},"references-count":27,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,10,1]]}},"URL":"https:\/\/doi.org\/10.1214\/26-aos2655","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,1]]}}}