{
  "id": "math/0601122",
  "version": "v3",
  "published": "2006-01-06T16:06:05.000Z",
  "updated": "2008-04-02T06:01:59.000Z",
  "title": "Navigation on a Poisson point process",
  "authors": [
    "Charles Bordenave"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/07-AAP472 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2008, Vol. 18, No. 2, 708-746",
  "doi": "10.1214/07-AAP472",
  "categories": [
    "math.PR"
  ],
  "abstract": "On a locally finite point set, a navigation defines a path through the point set from one point to another. The set of paths leading to a given point defines a tree known as the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on $\\mathbb{R}^d$. We examine the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small-world graphs where new results are established.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-04-02T06:01:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "05C05",
      "90C27",
      "60G55"
    ],
    "keywords": [
      "poisson point process",
      "navigation tree",
      "locally finite point set",
      "local weak convergence",
      "point defines"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1122B"
    }
  }
}