{
  "id": "1604.03993",
  "version": "v1",
  "published": "2016-04-13T22:52:05.000Z",
  "updated": "2016-04-13T22:52:05.000Z",
  "title": "Consistency of modularity clustering on random geometric graphs",
  "authors": [
    "Erik Davis",
    "Sunder Sethuraman"
  ],
  "comment": "4 figures, 55 pages",
  "categories": [
    "math.PR",
    "math.OC",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We consider a large class of random geometric graphs constructed from samples $\\mathcal{X}_n = \\{X_1,X_2,\\ldots,X_n\\}$ of independent, identically distributed observations of an underlying probability measure $\\nu$ on a bounded domain $D\\subset \\mathbb{R}^d$. The popular `modularity' clustering method specifies a partition $\\mathcal{U}_n$ of the set $\\mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $\\nu$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $\\mathcal{U}_n$ converge in a certain sense to a continuum partition $\\mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvin's shape optimization problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-13T22:52:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "62G20",
      "05C82",
      "49J55",
      "49J45",
      "68R10"
    ],
    "keywords": [
      "random geometric graphs",
      "modularity clustering",
      "consistency",
      "kelvins shape optimization problem",
      "discrete optimal partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160403993D"
    }
  }
}