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Pseudofractal Scale-free Web

S. N. Dorogovtsev·A. V. Goltsev·J. F. F. Mendes
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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 $γ=1+\ln3/\ln2$. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with $γ$ 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 $2+γ$.

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S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes. 2001-12-08. Pseudofractal Scale-free Web. https://doi.org/10.1103/physreve.65.066122

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