Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2007-05-10
Phys. Rev. E 75, 066110 (2007)
Physics
Condensed Matter
Disordered Systems and Neural Networks
10 pages, 5 figures
Scientific paper
10.1103/PhysRevE.75.066110
We propose a numerical method to evaluate the upper critical dimension $d_c$ of random percolation clusters in Erd\H{o}s-R\'{e}nyi networks and in scale-free networks with degree distribution ${\cal P}(k) \sim k^{-\lambda}$, where $k$ is the degree of a node and $\lambda$ is the broadness of the degree distribution. Our results report the theoretical prediction, $d_c = 2(\lambda - 1)/(\lambda - 3)$ for scale-free networks with $3 < \lambda < 4$ and $d_c = 6$ for Erd\H{o}s-R\'{e}nyi networks and scale-free networks with $\lambda > 4$. When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain $d_c = 6$ for all $\lambda > 2$. Our method also yields a better numerical evaluation of the critical percolation threshold, $p_c$, for scale-free networks. Our results suggest that the finite size effects increases when $\lambda$ approaches 3 from above.
Braunstein Lidia A.
Cohen Reuven
Havlin Shlomo
Lagorio Cecilia
Stanley Eugene H.
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