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

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 $\mathcal{P}(k)\sim k^{-\lambda }$, where $k$ is the degree of a node and $\lambda$ is the broadness of the degree distribution. Our results report the theoretical prediction, $d_c = 2(\lambda - 1)/(\lambda - 3)$forscale-freenetworkswith$3 < \lambda < 4$and$d_c = 6$forErd\stackrel{˝}{o}s-R\acute{e}nyinetworksandscale-freenetworkswith$\lambda > 4$. 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.