{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T09:15:31Z","timestamp":1790846131155,"version":"4.1.0"},"reference-count":21,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,3,16]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Asymmetric relational data are becoming increasingly prevalent in diverse fields, underscoring the need for developing directed network models to address the complex challenges posed by the unique structure of such data. Unlike undirected models, directed models can capture reciprocity, the tendency of nodes to form mutual links. This work addresses a fundamental question: what is the effective sample size for modelling reciprocity? We examine this question by analysing the Bernoulli model with reciprocity, allowing for varying sparsity levels between non-reciprocal and reciprocal effects. We then extend this framework to a model that incorporates node-specific heterogeneity and link-specific reciprocity using covariates. Our findings reveal the intriguing interplay between non-reciprocal and reciprocal effects in sparse networks. We propose a straightforward inference procedure based on maximum likelihood estimation that operates without prior knowledge of sparsity levels, whether covariates are included or not.<\/jats:p>","DOI":"10.1093\/biomet\/asaf035","type":"journal-article","created":{"date-parts":[[2025,6,2]],"date-time":"2025-06-02T14:20:26Z","timestamp":1748874026000},"source":"Crossref","is-referenced-by-count":1,"title":["Modelling directed networks with reciprocity"],"prefix":"10.1093","volume":"112","author":[{"given":"Rui","family":"Feng","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Warwick , Coventry CV4 7AL,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chenlei","family":"Leng","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Warwick , Coventry CV4 7AL,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,6,2]]},"reference":[{"key":"2025062913170289800_asaf035-B1","doi-asserted-by":"crossref","first-page":"21068","DOI":"10.1073\/pnas.0907096106","article-title":"A nonparametric view of network models and Newman\u2013Girvan and other modularities","volume":"106","author":"Bickel","year":"2009","journal-title":"Proc. Nat. Acad. Sci."},{"key":"2025062913170289800_asaf035-B2","doi-asserted-by":"crossref","first-page":"1400","DOI":"10.1214\/10-AAP728","article-title":"Random graphs with a given degree sequence","volume":"21","author":"Chatterjee","year":"2011","journal-title":"Ann. Appl. Prob"},{"key":"2025062913170289800_asaf035-B3","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1111\/rssb.12444","article-title":"Analysis of networks via the sparse $ \\beta $-model","volume":"83","author":"Chen","year":"2021","journal-title":"J. R. Statist. Soc. B"},{"key":"2025062913170289800_asaf035-B4","first-page":"290","article-title":"On random graphs","volume":"6","author":"Erd\u0151s","year":"1959","journal-title":"Publ. Math"},{"key":"2025062913170289800_asaf035-B5","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Publ. Math. Inst. Hung. Acad. Sci"},{"key":"2025062913170289800_asaf035-B6","doi-asserted-by":"crossref","first-page":"268701","DOI":"10.1103\/PhysRevLett.93.268701","article-title":"Patterns of link reciprocity in directed networks","volume":"93","author":"Garlaschelli","year":"2004","journal-title":"Phys. Rev. Lett"},{"key":"2025062913170289800_asaf035-B7","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":"2025062913170289800_asaf035-B8","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":"2025062913170289800_asaf035-B9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/01621459.1981.10477598","article-title":"An exponential family of probability distributions for directed graphs","volume":"76","author":"Holland","year":"1981","journal-title":"J. Am. Statist. Assoc"},{"key":"2025062913170289800_asaf035-B10","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1214\/14-STS502","article-title":"On the question of effective sample size in network modeling: An asymptotic inquiry","volume":"30","author":"Krivitsky","year":"2015","journal-title":"Statist. Sci."},{"key":"2025062913170289800_asaf035-B11","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199242726.001.0001","volume-title":"The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership","author":"Lazega","year":"2001"},{"key":"2025062913170289800_asaf035-B12","first-page":"86","article-title":"Universal latent space model fitting for large networks with edge covariates","volume":"21","author":"Ma","year":"2020","journal-title":"J. Mach. Learn. Res"},{"key":"2025062913170289800_asaf035-B13","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1090\/conm\/516\/10180","article-title":"Algebraic statistics for a directed random graph model with reciprocation","volume":"516","author":"Petrovic","year":"2010","journal-title":"Contemp. Math"},{"key":"2025062913170289800_asaf035-B14","author":"Qu","year":"2025"},{"key":"2025062913170289800_asaf035-B15","author":"Rinaldo","year":"2010"},{"key":"2025062913170289800_asaf035-B16","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.socnet.2006.08.002","article-title":"An introduction to exponential random graph ($ p^{\\ast} $) models for social networks","volume":"29","author":"Robins","year":"2007","journal-title":"Social Networks"},{"key":"2025062913170289800_asaf035-B17","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1046\/j.0039-0402.2003.00258.x","article-title":"$ p_{2} $: A random effects model with covariates for directed graphs","volume":"58","author":"Van Duijn","year":"2004","journal-title":"Statist. Neer"},{"key":"2025062913170289800_asaf035-B18","author":"Wolfe","year":"2013"},{"key":"2025062913170289800_asaf035-B19","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1080\/01621459.2018.1448829","article-title":"Statistical inference in a directed network model with covariates","volume":"114","author":"Yan","year":"2019","journal-title":"J. Am. Statist. Assoc"},{"key":"2025062913170289800_asaf035-B20","doi-asserted-by":"crossref","first-page":"255","DOI":"10.4310\/SII.2015.v8.n3.a1","article-title":"A simulation study of the $ p_{1} $ model for directed random graphs","volume":"8","author":"Yan","year":"2015","journal-title":"Statist. Interface"},{"key":"2025062913170289800_asaf035-B21","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1214\/15-AOS1343","article-title":"Asymptotics in directed exponential random graph models with an increasing bi-degree sequence","volume":"44","author":"Yan","year":"2016","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\/asaf035\/63417429\/asaf035.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/2\/asaf035\/63417429\/asaf035.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/2\/asaf035\/63417429\/asaf035.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,29]],"date-time":"2025-06-29T17:17:12Z","timestamp":1751217432000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asaf035\/8155126"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,3,16]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaf035","relation":{},"ISSN":["1464-3510"],"issn-type":[{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025]]},"published":{"date-parts":[[2025]]},"article-number":"asaf035"}}