English

Pseudofractal Scale-free Web

Statistical Mechanics 2009-11-07 v1

Abstract

We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent γ=1+ln3/ln2\gamma=1+\ln3/\ln2. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with γ\gamma in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network (lnN1\ln N \gg 1) the distribution tends to a Gaussian of width lnN\sim \sqrt{\ln N} centered at ˉlnN\bar{\ell} \sim \ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+γ2+\gamma.

Keywords

Cite

@article{arxiv.cond-mat/0112143,
  title  = {Pseudofractal Scale-free Web},
  author = {S. N. Dorogovtsev and A. V. Goltsev and J. F. F. Mendes},
  journal= {arXiv preprint arXiv:cond-mat/0112143},
  year   = {2009}
}

Comments

5 pages, 3 figures