Numerical evaluation of the upper critical dimension of percolation in scale-free networks

Abstract

We propose a numerical method to evaluate the upper critical dimension $d_c$ of random percolation clusters in Erdős-Rényi networks and in scale-free networks with degree distribution $\mathcal{P}(k)\sim k^{-\lambda }$, 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 Erdős-Rényi 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.

arXiv comment10 pages, 5 figures

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