{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T05:14:35Z","timestamp":1780550075137,"version":"3.54.1"},"reference-count":37,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2021,11,11]],"date-time":"2021-11-11T00:00:00Z","timestamp":1636588800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,8,24]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We consider local linear estimation of the graphon function, which determines probabilities of pairwise edges between nodes in an unlabelled network. Real-world networks are typically characterized by node heterogeneity, with different nodes exhibiting different degrees of interaction. Existing approaches to graphon estimation are limited to local constant approximations, and are not designed to estimate heterogeneity across the full network. In this paper, we show how continuous node covariates can be employed to estimate heterogeneity in the network via a local linear graphon estimator. We derive the bias and variance of an oracle-based local linear graphon estimator, and thus obtain the mean integrated squared error optimal bandwidth rule. We also provide a plug-in bandwidth selection procedure that makes local linear estimation for unlabelled networks practically feasible. The finite-sample performance of our approach is investigated in a simulation study, and the method is applied to a school friendship network and an email network to illustrate its advantages over existing methods.<\/jats:p>","DOI":"10.1093\/biomet\/asab057","type":"journal-article","created":{"date-parts":[[2021,11,5]],"date-time":"2021-11-05T20:16:02Z","timestamp":1636143362000},"page":"721-734","source":"Crossref","is-referenced-by-count":6,"title":["Local linear graphon estimation using covariates"],"prefix":"10.1093","volume":"109","author":[{"given":"S","family":"Chandna","sequence":"first","affiliation":[{"name":"Department of Economics, Mathematics and Statistics, Birkbeck, University of London , Malet Street, London WC1E 7HX, U.K"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S C","family":"Olhede","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne , Station 8, 1015 Lausanne, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P J","family":"Wolfe","sequence":"additional","affiliation":[{"name":"Department of Statistics, Purdue University , 150 N. 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