Percolating critical window for correlated scale-free networks

Abstract

In scale-free networks, correlations between degrees could be induced by the forbidding of multiple connections between hubs. In particular, for scale-free networks with $2<\lambda <3$, where there is a natural cut-off on degrees, such correlations naturally arise. This kind of correlation was usually ignored when considering the critical percolation in these networks. To quantify its effect on the behavior of critical percolation, we consider a scale-dependent truncation $N^{\zeta }$, which characterizes the strength of the degree-degree correlations, and a degree truncation $N^{\kappa }$, which characterizes the scale influence. Based on this model, we first analyze the critical behavior of percolation phase transitions using message passing methods. Our finding suggests that in scale-free networks with $2<\lambda <3$, the critical window is $|q_c\left(N\right)-q_c\left(\infty \right)|\sim N^{-min(\zeta /2,\kappa )\times (3-\lambda )}$, where $q_c$ denotes the critical occupied probability of sites. This indicates the degree-degree correlation would modify the universal class of percolation transition, and this phenomenon is absent in scale-free networks with $3<\lambda <4$. These analytical results are then inspected by numerical simulations.