{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:00:51Z","timestamp":1760148051098,"version":"build-2065373602"},"reference-count":70,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T00:00:00Z","timestamp":1679529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific research start-up fund of CUMT","award":["102520253","bJSSCBS20211218"],"award-info":[{"award-number":["102520253","bJSSCBS20211218"]}]},{"name":"High-level personal project of Jiangsu Province","award":["102520253","bJSSCBS20211218"],"award-info":[{"award-number":["102520253","bJSSCBS20211218"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Community detection in weighted networks has been a popular topic in recent years. However, while there exist several flexible methods for estimating communities in weighted networks, these methods usually assume that the number of communities is known. It is usually unclear how to determine the exact number of communities one should use. Here, to estimate the number of communities for weighted networks generated from arbitrary distribution under the degree-corrected distribution-free model, we propose one approach that combines weighted modularity with spectral clustering. This approach allows a weighted network to have negative edge weights and it also works for signed networks. We compare the proposed method to several existing methods and show that our method is more accurate for estimating the number of communities both numerically and empirically.<\/jats:p>","DOI":"10.3390\/e25040551","type":"journal-article","created":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T03:16:46Z","timestamp":1679627806000},"page":"551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Estimating the Number of Communities in Weighted Networks"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7496-5022","authenticated-orcid":false,"given":"Huan","family":"Qing","sequence":"first","affiliation":[{"name":"School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. 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