{
  "id": "1401.0409",
  "version": "v2",
  "published": "2014-01-02T10:43:54.000Z",
  "updated": "2014-09-26T17:47:32.000Z",
  "title": "Inhomogeneous Long-Range Percolation for Real-Life Network Modeling",
  "authors": [
    "Rajat Subhra Hazra",
    "Mario V. Wüthrich"
  ],
  "comment": "19 pages, new version. Substantially modified version",
  "categories": [
    "math.PR"
  ],
  "abstract": "The study of random graphs has become very popular for real-life network modeling such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice $\\mathbb Z^d$, $d\\ge1$, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model various geometric properties such as the percolation behavior, the degree distribution and graph distances have been analyzed. In the present paper we complement the picture about graph distances. Moreover, we prove continuity of the percolation probability in the phase transition point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-02T10:43:54.000Z",
      "title": "Continuity of the percolation probability and chemical distances in inhomogeneous long-range percolation",
      "abstract": "Inhomogeneous long-range percolation on the lattice $\\mathbb Z^d$ was introduced in Deijfen et al. (2013) as an extension of the homogeneous long-range percolation model. The inhomogeneous long-range percolation model assigns i.i.d. weights $W_x$ to each vertex $x\\in \\mathbb Z^d$. Conditionally on these weights, an edge between vertices $x$ and $y$ is occupied with probability $p_{xy}=1-\\exp(-\\lambda W_xW_y|x-y|^{-\\alpha})$, independently of all other edges. Deijfen et al. (2013) provides the phase transition picture for the existence of an infinite component of occupied edges. In the present paper we complement this phase transition picture by proving that the percolation probability (as a function of $\\lambda$)is continuous for $\\alpha\\in(d, 2d)$ and, therefore, there is no infinite component at criticality. Moreover, we complement the picture of Deijfen et al. (2013) about chemical distances in the infinite component.",
      "comment": "24 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-26T17:47:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "percolation probability",
      "chemical distances",
      "phase transition picture",
      "infinite component",
      "continuity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.0409D"
    }
  }
}