{"title":"随机图的平均杰卡尔指数","authors":"Qunqiang Feng, Shuai Guo, Zhishui Hu","doi":"10.1017/jpr.2023.112","DOIUrl":null,"url":null,"abstract":"The asymptotic behavior of the Jaccard index in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>), the classical Erdös–Rényi random graph model, is studied as <jats:italic>n</jats:italic> goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>) is shown to be asymptotically normal under an additional mild condition that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223001122_inline1.png\" /> <jats:tex-math> $np\\to\\infty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900223001122_inline2.png\" /> <jats:tex-math> $n^2(1-p)\\to\\infty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"12 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Average Jaccard index of random graphs\",\"authors\":\"Qunqiang Feng, Shuai Guo, Zhishui Hu\",\"doi\":\"10.1017/jpr.2023.112\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The asymptotic behavior of the Jaccard index in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>), the classical Erdös–Rényi random graph model, is studied as <jats:italic>n</jats:italic> goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in <jats:italic>G</jats:italic>(<jats:italic>n</jats:italic>, <jats:italic>p</jats:italic>) is shown to be asymptotically normal under an additional mild condition that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900223001122_inline1.png\\\" /> <jats:tex-math> $np\\\\to\\\\infty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0021900223001122_inline2.png\\\" /> <jats:tex-math> $n^2(1-p)\\\\to\\\\infty$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":50256,\"journal\":{\"name\":\"Journal of Applied Probability\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-02-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Applied Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/jpr.2023.112\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2023.112","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
The asymptotic behavior of the Jaccard index in G(n, p), the classical Erdös–Rényi random graph model, is studied as n goes to infinity. We first derive the asymptotic distribution of the Jaccard index of any pair of distinct vertices, as well as the first two moments of this index. Then the average of the Jaccard indices over all vertex pairs in G(n, p) is shown to be asymptotically normal under an additional mild condition that $np\to\infty$ and $n^2(1-p)\to\infty$ .
期刊介绍：
Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used.
A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
