{
  "id": "cond-mat/0201083",
  "version": "v1",
  "published": "2002-01-07T21:36:07.000Z",
  "updated": "2002-01-07T21:36:07.000Z",
  "title": "Shortest paths on systems with power-law distributed long-range connections",
  "authors": [
    "Cristian F. Moukarzel",
    "Marcio Argollo de Menezes"
  ],
  "comment": "10 pages, 10 figures, revtex4. Submitted to PRE",
  "doi": "10.1103/PhysRevE.65.056709",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn"
  ],
  "abstract": "We discuss shortest-path lengths $\\ell(r)$ on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to $P_l \\sim l^{-\\xpn}$. Using rescaling arguments and numerical simulation on systems of up to $10^7$ sites, we show that a characteristic length $\\xi$ exists such that $\\ell(r) \\sim r$ for $r<\\xi$ but $\\ell(r) \\sim r^{\\theta_s(\\xpn)}$ for $r>>\\xi$. For small p we find that the shortest-path length satisfies the scaling relation $\\ell(r,\\xpn,p)/\\xi = f(\\xpn,r/\\xi)$. Three regions with different asymptotic behaviors are found, respectively: a) $\\xpn>2$ where $\\theta_s=1$, b) $1<\\xpn<2$ where $0<\\theta_s(\\xpn)<1/2$ and, c) $\\xpn<1$ where $\\ell(r)$ behaves logarithmically, i.e. $\\theta_s=0$. The characteristic length $\\xi$ is of the form $\\xi \\sim p^{-\\nu}$ with $\\nu=1/(2-\\xpn)$ in region b), but depends on L as well in region c). A directed model of shortest-paths is solved and compared with numerical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-01-07T21:36:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "power-law distributed long-range connections",
      "shortest paths",
      "characteristic length",
      "pl randomly located long-range links",
      "shortest-path length satisfies"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}