{
  "id": "0908.3436",
  "version": "v1",
  "published": "2009-08-24T15:21:52.000Z",
  "updated": "2009-08-24T15:21:52.000Z",
  "title": "Rank-based attachment leads to power law graphs",
  "authors": [
    "Jeannette Janssen",
    "Pawel Pralat"
  ],
  "journal": "SIAM Journal of Discrete Math 24, 2010, pp. 420--440",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-24T15:21:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C07"
    ],
    "keywords": [
      "rank-based attachment",
      "power law graphs",
      "power law degree distribution",
      "initial rank",
      "graph generation models"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.3436J"
    }
  }
}