{
  "id": "1703.06328",
  "version": "v1",
  "published": "2017-03-18T18:16:57.000Z",
  "updated": "2017-03-18T18:16:57.000Z",
  "title": "Functional Central Limit Theorem For Susceptible-Infected Process On Configuration Model Graphs",
  "authors": [
    "Wasiur R. KhudaBukhsh",
    "Casper Woroszylo",
    "Grzegorz A. Rempała",
    "Heinz Koeppl"
  ],
  "comment": "52 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a stochastic compartmental susceptible-infected (SI) epidemic process on a configuration model random graph with a given degree distribution over a finite time interval $[0,T],$ for some $ T>0$. In this setting, we split the population of graph nodes into two compartments, namely, $S$ and $I$, denoting the susceptible and infected nodes, respectively. In addition to the sizes of these two compartments, we study counts of $SI$-edges (those connecting a susceptible and an infected node) and $SS$-edges (those connecting two susceptible nodes). We describe the dynamical process in terms of these counts and present a functional central limit theorem (FCLT) for them, a scaling limit of the dynamical process as $n$, the number of nodes in the random graph, grows to infinity. To be precise, we show that these counts, when appropriately scaled, converge weakly to a continuous Gaussian vector martingale process the usual Skorohod space of real 3-dimensional vector-valued \\cadlag\\, functions on $[0,T]$ endowed with the Skorohod topology. We assume certain technical requirements for this purpose. We discuss applications of our FCLT in percolation theory (from a non-equilibrium statistical mechanics point of view), and in computer science in the context of spread of computer viruses. We also provide simulation results for some common degree distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-18T18:16:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60F05",
      "92D30",
      "05C80"
    ],
    "keywords": [
      "functional central limit theorem",
      "configuration model graphs",
      "susceptible-infected process",
      "configuration model random graph",
      "continuous gaussian vector martingale process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}