Numerical assessment of the percolation threshold using complement networks

Abstract

Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold $p_c$ and its model-predicted value ${\pi }_c$. Here we show the existence of an empirical linear relation between $p_c$ and ${\pi }_c$ across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of ${\pi }_c$. We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, $p_c$, and that of its complement, ${\bar{p}}_c$.