{
  "id": "1910.13958",
  "version": "v1",
  "published": "2019-10-30T16:17:19.000Z",
  "updated": "2019-10-30T16:17:19.000Z",
  "title": "Critical value asymptotics for the contact process on random graphs",
  "authors": [
    "Danny Nam",
    "Oanh Nguyen",
    "Allan Sly"
  ],
  "comment": "57 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Recent progress in the study of the contact process [2] has verified that the extinction-survival threshold $\\lambda_1$ on a Galton-Watson tree is strictly positive if and only if the offspring distribution $\\xi$ has an exponential tail. In this paper, we derive the first-order asymptotics of $\\lambda_1$ for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if $\\xi$ is appropriately concentrated around its mean, we demonstrate that $\\lambda_1(\\xi) \\sim 1/\\mathbb{E} \\xi$ as $\\mathbb{E}\\xi\\rightarrow \\infty$, which matches with the known asymptotics on the $d$-regular trees. The same result for the short-long survival threshold on the Erd\\H{o}s-R\\'enyi and other random graphs are shown as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-30T16:17:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "contact process",
      "random graphs",
      "critical value asymptotics",
      "galton-watson tree",
      "short-long survival threshold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}