{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T05:10:52Z","timestamp":1788844252605,"version":"build-2803163510"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,5]],"date-time":"2026-09-05T00:00:00Z","timestamp":1788566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100017596","name":"Natural Science Basic Research Program of Shaanxi Province","doi-asserted-by":"crossref","award":["2026JC-YBQN-0974"],"award-info":[{"award-number":["2026JC-YBQN-0974"]}],"id":[{"id":"10.13039\/501100017596","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Humanities and Social Sciences Fund of the Ministry of Education of China","award":["23YJC630231"],"award-info":[{"award-number":["23YJC630231"]}]},{"DOI":"10.13039\/501100013794","name":"Shanxi Provincial Education Department","doi-asserted-by":"crossref","award":["23JT032"],"award-info":[{"award-number":["23JT032"]}],"id":[{"id":"10.13039\/501100013794","id-type":"DOI","asserted-by":"crossref"}]},{"award":["23JT032"],"award-info":[{"award-number":["23JT032"]}],"id":[{"id":"https:\/\/ror.org\/05r16bb17","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72474175"],"award-info":[{"award-number":["72474175"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["72474175"],"award-info":[{"award-number":["72474175"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Data obtained from experiments and surveys in human social systems are inevitably influenced by systematic measurement errors, and network data are no exception. Despite the prevalence of error in social network data, current research often lacks rigorous estimation of its expected precision, which may lead to biased conclusions. Signed networks, which encode both positive and negative relationships, constitute an important component of network science, and conducting measurement error analysis on them can substantially enhance the accuracy of social network analysis. This paper proposes a set of error measurement tools based on the Expectation-Maximization (EM) algorithm, specifically designed to estimate errors in signed network data. We extend traditional experimental error estimation to the network domain, derive a general error estimation method for signed networks, and validate its scientific validity and practical utility through extensive simulation experiments on both synthetic and real-world networks. The experiments reveal that network density and the ratio of positive to negative edges significantly influence the posterior probability distribution of the adjacency matrix. Specifically, as density increases, edge estimation accuracy exhibits a U-shaped trend, and the proportion of negative edges shows a nonlinear relationship with accuracy. The proposed method is applicable to repeatedly measured signed networks and provides a reliable framework for reconstructing network structures as faithfully as possible.<\/jats:p>","DOI":"10.3390\/e28090993","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T16:30:02Z","timestamp":1788798602000},"page":"993","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Error Estimation of Signed Networks Based on Expectation-Maximization Algorithm"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4173-0937","authenticated-orcid":false,"given":"Ruochen","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Economics and Management, Xi\u2019an Shiyou University, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"}]},{"given":"Zijie","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Public Policy and Administration, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5676-6812","authenticated-orcid":false,"given":"Jiarui","family":"Fan","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Xi\u2019an University, Xi\u2019an 710065, China"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"validation","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"data-curation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"supervision","vocabulary":"credit"},{"role":"project-administration","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Knoke, D., and Yang, S. 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