{
  "id": "cond-mat/0605587",
  "version": "v2",
  "published": "2006-05-24T04:32:50.000Z",
  "updated": "2008-04-29T08:49:30.000Z",
  "title": "Self-similarity in Fractal and Non-fractal Networks",
  "authors": [
    "J. S. Kim",
    "B. Kahng",
    "D. Kim",
    "K. -I. Goh"
  ],
  "comment": "15 pages, 8 figures",
  "journal": "Journal of Korean Physical Society 52, 350 (2008)",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn"
  ],
  "abstract": "We study the origin of scale invariance (SI) of the degree distribution in scale-free (SF) networks with a degree exponent $\\gamma$ under coarse graining. A varying number of vertices belonging to a community or a box in a fractal analysis is grouped into a supernode, where the box mass $M$ follows a power-law distribution, $P_m(M)\\sim M^{-\\eta}$. The renormalized degree $k^{\\prime}$ of a supernode scales with its box mass $M$ as $k^{\\prime} \\sim M^{\\theta}$. The two exponents $\\eta$ and $\\theta$ can be nontrivial as $\\eta \\ne \\gamma$ and $\\theta <1$. They act as relevant parameters in determining the self-similarity, i.e., the SI of the degree distribution, as follows: The self-similarity appears either when $\\gamma \\le \\eta$ or under the condition $\\theta=(\\eta-1)/(\\gamma-1)$ when $\\gamma> \\eta$, irrespective of whether the original SF network is fractal or non-fractal. Thus, fractality and self-similarity are disparate notions in SF networks.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-29T08:49:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-fractal networks",
      "degree distribution",
      "box mass",
      "original sf network",
      "scale invariance"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006cond.mat..5587K"
    }
  }
}