{
  "id": "2205.03551",
  "version": "v1",
  "published": "2022-05-07T04:48:48.000Z",
  "updated": "2022-05-07T04:48:48.000Z",
  "title": "Subcritical epidemics on random graphs",
  "authors": [
    "Oanh Nguyen",
    "Allan Sly"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the contact process on random graphs with low infection rate $\\lambda$. For random $d$-regular graphs, it is known that the survival time is $O(\\log n)$ below the critical $\\lambda_c$. By contrast, on the Erd\\H{o}s-R\\'enyi random graphs $\\mathcal G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{\\lambda^{2+o(1)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a shaper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-07T04:48:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graphs",
      "subcritical epidemics",
      "contact process",
      "rare high-degree vertices result",
      "low infection rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}