{
  "id": "2408.03218",
  "version": "v1",
  "published": "2024-08-06T14:23:55.000Z",
  "updated": "2024-08-06T14:23:55.000Z",
  "title": "Limit theorems for the number of crossings and stress in projections of the random geometric graph",
  "authors": [
    "Hanna Döring",
    "Lianne de Jonge"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\\subset \\mathbb{R}^d$, $d\\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-06T14:23:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60D05"
    ],
    "keywords": [
      "multivariate central limit theorem",
      "random geometric graph converges",
      "projections",
      "finite value",
      "compact convex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}