{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T15:13:51Z","timestamp":1791558831181,"version":"4.3.4"},"reference-count":50,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T00:00:00Z","timestamp":1708732800000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Community detection is a crucial task in network analysis that can be significantly improved by incorporating subject-level information, ie, covariates. Existing methods have shown the effectiveness of using covariates on the low-degree nodes, but rarely discuss the case where communities have significantly different density levels, ie, multiscale networks. In this paper, we introduce a novel method that addresses this challenge by constructing network-adjusted covariates, which leverage the network connections and covariates with a node-specific weight for each node. This weight can be calculated without tuning parameters. We present novel theoretical results on the strong consistency of our method under degree-corrected stochastic blockmodels with covariates, even in the presence of misspecification and multiple sparse communities. Additionally, we establish a general lower bound for the community detection problem when both the network and covariates are present, and it shows that our method is optimal for connection intensity up to a constant factor. Our method outperforms existing approaches in simulations and a LastFM app user network. We then compare our method with others on a statistics publication citation network where 30% of nodes are isolated, and our method produces reasonable and balanced results. Our method is implemented in the R package NAC.<\/jats:p>","DOI":"10.1093\/biomet\/asae011","type":"journal-article","created":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T13:47:51Z","timestamp":1708782471000},"page":"1221-1240","source":"Crossref","is-referenced-by-count":10,"title":["Network-adjusted covariates for community detection"],"prefix":"10.1093","volume":"111","author":[{"given":"Y","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, Southern Methodist University , 3225 Daniel Avenue , Dallas, Texas 75205,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science, National University of Singapore , S16 Science Drive 2 , 117546","place":["Singapore"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,2,23]]},"reference":[{"key":"2024112207520870800_asae011-B1","first-page":"6446","article-title":"Community detection and stochastic block models: recent developments","volume":"18","author":"Abbe","year":"2017","journal-title":"J. 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