{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T09:15:33Z","timestamp":1790846133396,"version":"4.1.0"},"reference-count":53,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2023,2,24]],"date-time":"2023-02-24T00:00:00Z","timestamp":1677196800000},"content-version":"vor","delay-in-days":536,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/100012338","name":"Alan Turing Institute","doi-asserted-by":"publisher","award":["EP\/N510129\/1"],"award-info":[{"award-number":["EP\/N510129\/1"]}],"id":[{"id":"10.13039\/100012338","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,11,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Data in the form of networks are increasingly available in a variety of areas, yet statistical models allowing for parameter estimates with desirable statistical properties for sparse networks remain scarce. To address this, we propose the Sparse \u03b2-Model (S\u03b2M), a new network model that interpolates the celebrated Erd\u0151s\u2013R\u00e9nyi model and the \u03b2-model that assigns one different parameter to each node. By a novel reparameterization of the \u03b2-model to distinguish global and local parameters, our S\u03b2M can drastically reduce the dimensionality of the \u03b2-model by requiring some of the local parameters to be zero. We derive the asymptotic distribution of the maximum likelihood estimator of the S\u03b2M when the support of the parameter vector is known. When the support is unknown, we formulate a penalized likelihood approach with the \u21130-penalty. Remarkably, we show via a monotonicity lemma that the seemingly combinatorial computational problem due to the \u21130-penalty can be overcome by assigning non-zero parameters to those nodes with the largest degrees. We further show that a \u03b2-min condition guarantees our method to identify the true model and provide excess risk bounds for the estimated parameters. The estimation procedure enjoys good finite sample properties as shown by simulation studies. The usefulness of the S\u03b2M is further illustrated via the analysis of a microfinance take-up example.<\/jats:p>","DOI":"10.1111\/rssb.12444","type":"journal-article","created":{"date-parts":[[2021,9,6]],"date-time":"2021-09-06T14:04:35Z","timestamp":1630937075000},"page":"887-910","source":"Crossref","is-referenced-by-count":24,"title":["Analysis of Networks via the Sparse<i>\u03b2<\/i>-model"],"prefix":"10.1093","volume":"83","author":[{"given":"Mingli","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Economics , , Coventry , UK"},{"name":"University of Warwick , , Coventry , UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kengo","family":"Kato","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science , , Ithaca, New York , USA"},{"name":"Cornell University , , Ithaca, New York , 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