Pseudofractal Scale-free Web

Details Pseudofractal Scale-free Web Pseudofractal Scale-free Web

2001-12-08

arxiv.org/abs/cond-mat/0112143v1

Phys. Rev. E 65, 066122 (2002)

Physics

Condensed Matter

Statistical Mechanics

5 pages, 3 figures

Scientific paper

10.1103/PhysRevE.65.066122

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 $\gamma=1+\ln3/\ln2$. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with $\gamma$ 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 ($\ln N \gg 1$) the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\bar{\ell} \sim \ln N$. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\gamma$.

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