{
  "id": "2311.09804",
  "version": "v1",
  "published": "2023-11-16T11:29:03.000Z",
  "updated": "2023-11-16T11:29:03.000Z",
  "title": "Average Jaccard Index of Random Graphs",
  "authors": [
    "Qunqiang Feng",
    "Shuai Guo",
    "Zhishui Hu"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "The asymptotic behavior of the Jaccard index in $G(n,p)$, the classical Erd\\\"{o}s-R\\'{e}nyi random graphs model, is studied in this paper, 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 $n^2(1-p)\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-16T11:29:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "average jaccard index",
      "random graphs model",
      "additional mild condition",
      "vertex pairs",
      "asymptotic behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}