{
  "id": "0809.1143",
  "version": "v1",
  "published": "2008-09-06T07:30:51.000Z",
  "updated": "2008-09-06T07:30:51.000Z",
  "title": "Number of Edges in Random Intersection Graph on Surface of a Sphere",
  "authors": [
    "Bhupendra gupta"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we consider `$N$'spherical caps of area $4\\pi p$ were uniformly distributed over the surface of a unit sphere. We study the random intersection graph $G_N$ constructed by these caps. We prove that for $p = \\frac{c}{N^{\\al}},\\:c >0$ and $\\al >2,$ the number of edges in graph $G_N$ follow the Poisson distribution. Also we derive the strong law results for the number of isolated vertices in $G_N$: for $p = \\frac{c}{N^{\\al}},\\:c >0$ for $\\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are atleast $N/2$ edges in $G_N$ and for $\\al >3,$ every vertex in $G_N$ is isolated i.e., there is no edge in edge set $\\cE_N.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-06T07:30:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random intersection graph",
      "strong law results",
      "unit sphere",
      "edge set",
      "poisson distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.1143G"
    }
  }
}