{
  "id": "1012.4336",
  "version": "v1",
  "published": "2010-12-20T14:08:43.000Z",
  "updated": "2010-12-20T14:08:43.000Z",
  "title": "Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks",
  "authors": [
    "G. J. Baxter",
    "S. N. Dorogovtsev",
    "A. V. Goltsev",
    "J. F. F. Mendes"
  ],
  "comment": "10 pages, 4 figures",
  "journal": "Physical Review E 83, 051134 (2011)",
  "doi": "10.1103/PhysRevE.83.051134",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \\geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-$k$-core process: the giant heterogeneous-$k$-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-$k$-core and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$ when the degree distribution tends to a power law $P(q) \\sim q^{-\\gamma}$ with $\\gamma < 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-20T14:08:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "64.60.aq",
      "64.60.ah",
      "05.10.-a",
      "05.70.Fh"
    ],
    "keywords": [
      "complex networks",
      "heterogeneous-k-core",
      "degree distribution tends",
      "bootstrap percolation transitions",
      "ordinary bootstrap percolation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "year": 2011,
      "month": "May",
      "volume": 83,
      "number": 5,
      "pages": "051134"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011PhRvE..83e1134B"
    }
  }
}