{
  "id": "2409.11615",
  "version": "v1",
  "published": "2024-09-18T00:41:47.000Z",
  "updated": "2024-09-18T00:41:47.000Z",
  "title": "The Moran process on a random graph",
  "authors": [
    "Alan Frieze",
    "Wesley Pegden"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We study the fixation probability for two versions of the Moran process on the random graph $G_{n,p}$ at the threshold for connectivity. The Moran process models the spread of a mutant population in a network. Throughtout the process there are vertices of two types, mutants and non-mutants. Mutants have fitness $s$ and non-mutants have fitness 1. The process starts with a unique individual mutant located at the vertex $v_0$. In the Birth-Death version of the process a random vertex is chosen proportional to its fitness and then changes the type of a random neighbor to its own. The process continues until the set of mutants $X$ is empty or $[n]$. In the Death-Birth version a uniform random vertex is chosen and then takes the type of a random neighbor, chosen according to fitness. The process again continues until the set of mutants $X$ is empty or $[n]$. The {\\em fixation probability} is the probability that the process ends with $X=\\emptyset$. We give asymptotically correct estimates of the fixation probability that depend on degree of $v_0$ and its neighbors.,",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-18T00:41:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random graph",
      "fixation probability",
      "random neighbor",
      "uniform random vertex",
      "moran process models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}