{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:57Z","timestamp":1760147037563,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T00:00:00Z","timestamp":1672704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"],"award-info":[{"award-number":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"]}]},{"name":"Humanities and Social Sciences Youth Foundation, Ministry of Education of the People\u2019s Republic of China","award":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"],"award-info":[{"award-number":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"]}]},{"name":"Scientific Research Program of Hubei Provincial Department of Education","award":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"],"award-info":[{"award-number":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"]}]},{"name":"Philosophy and Social Sciences Youth Foundation, Higher Education Institutions of Hubei Province","award":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"],"award-info":[{"award-number":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"]}]},{"name":"2021 Internal Scientific Research Fund Project of Wuhan Institute of Technology","award":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"],"award-info":[{"award-number":["72102171","61873108","21YJC630006","Q20211507","21Q087","K2021049"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.<\/jats:p>","DOI":"10.3390\/sym15010137","type":"journal-article","created":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T02:51:39Z","timestamp":1672714299000},"page":"137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Regularity Index of Uncertain Random Graph"],"prefix":"10.3390","volume":"15","author":[{"given":"Lin","family":"Chen","sequence":"first","affiliation":[{"name":"School of Management, Wuhan Institute of Technology, Wuhan 430205, China"}]},{"given":"Li","family":"Zeng","sequence":"additional","affiliation":[{"name":"School of Management, Wuhan Institute of Technology, Wuhan 430205, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7354-5034","authenticated-orcid":false,"given":"Jin","family":"Peng","sequence":"additional","affiliation":[{"name":"Institute of Uncertain Systems, Huanggang Normal University, Huanggang 438000, China"}]},{"given":"Junren","family":"Ming","sequence":"additional","affiliation":[{"name":"School of Management, Wuhan Institute of Technology, Wuhan 430205, China"}]},{"given":"Xianghui","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Management, Wuhan Institute of Technology, Wuhan 430205, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hu, X., Teng, J., Wu, W., Li, Y., and Sheng, Y. 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