{
  "id": "1310.5672",
  "version": "v2",
  "published": "2013-10-21T18:43:56.000Z",
  "updated": "2015-06-19T06:53:48.000Z",
  "title": "Degree distribution of shortest path trees and bias of network sampling algorithms",
  "authors": [
    "Shankar Bhamidi",
    "Jesse Goodman",
    "Remco van der Hofstad",
    "Júlia Komjáthy"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1036 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 4, 1780-1826",
  "doi": "10.1214/14-AAP1036",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random $r$-regular graphs for large $r$, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010) 1907-1965], [Combin. Probab. Comput. 20 (2011) 683-707], [Adv. in Appl. Probab. 42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-21T18:43:56.000Z",
      "abstract": "In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the power-law exponent of the degree distribution of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance) as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random $r$-regular graphs for large $r$, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean field model of distance. We use our results to explain an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra of analyzing the effect of attaching random edge lengths on the geometry of random network models.",
      "comment": "36 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-19T06:53:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "05C80",
      "90B15"
    ],
    "keywords": [
      "shortest path tree",
      "degree distribution",
      "network sampling algorithms",
      "random network models",
      "power law"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.5672B"
    }
  }
}