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

We propose numerical methods to evaluate the upper critical dimension dc of random percolation clusters in Erdos-Rényi networks and in scale-free networks with degree distribution P (k) ∼ k-λ, where k is the degree of a node and λ is the broadness of the degree distribution. Our results support the theoretical prediction, dc =2 (λ-1) (λ-3) for scale-free networks with 3<λ<4 and dc =6 for Erdos-Rényi networks and scale-free networks with λ>4. When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain dc =6 for all λ>2. Our method also yields a better numerical evaluation of the critical percolation threshold pc for scale-free networks. Our results suggest that the finite size effects increases when λ approaches 3 from above.