{
  "id": "1504.06840",
  "version": "v1",
  "published": "2015-04-26T15:58:22.000Z",
  "updated": "2015-04-26T15:58:22.000Z",
  "title": "Diameter and Stationary Distribution of Random $r$-out Digraphs",
  "authors": [
    "Louigi Addario-Berry",
    "Borja Balle",
    "Guillem Perarnau"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR",
    "cs.DM",
    "math.CO"
  ],
  "abstract": "Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\\{1,\\ldots,n\\}$. In this work, we establish that for every $r \\ge 2$, there exists $\\eta_r>0$ such that $\\text{diam}(D(n,r))=(1+\\eta_r+o(1))\\log_r{n}$. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\\pi_{\\max}$ and $\\pi_{\\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\\pi_{\\max} = n^{-1+o(1)}$ and $\\pi_{\\min}=n^{-(1+\\eta_r)+o(1)}$. Our proof shows that the vertices with $\\pi(v)$ near to $\\pi_{\\min}$ lie at the top of \"narrow, slippery towers\", such vertices are also responsible for increasing the diameter from $(1+o(1))\\log_r n$ to $(1+\\eta_r+o(1))\\log_r{n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-26T15:58:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stationary distribution",
      "simple random walk",
      "high probability",
      "minimum values",
      "asymptotic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150406840A"
    }
  }
}