Pseudofractal Scale-free Web

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 $\gamma =1+\ln 3/\ln 2$. 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 $2+\gamma$.

5 pages, 3 figures

References in corpus (5)

• Statistical mechanics of complex networks
• Evolution of networks
• Deterministic Scale-Free Networks
• Statistical ensemble of scale-free random graphs
• Comment on "Breakdown of the Internet under Intentional Attack"

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