Width of percolation transition in complex networks

Abstract

It is known that the critical probability for the percolation transition is not a sharp threshold, actually it is a region of non-zero width $\Delta p_c$ for systems of finite size. Here we present evidence that for complex networks $\Delta p_c\sim \frac{p_c}{l}$, where $l\sim N^{{\nu }_{opt}}$ is the average length of the percolation cluster, and $N$ is the number of nodes in the network. For Erdős-Rényi (ER) graphs ${\nu }_{opt}=1/3$, while for scale-free (SF) networks with a degree distribution $P(k)\sim k^{-\lambda }$ and $3<\lambda <4$, ${\nu }_{opt}=(\lambda -3)/(\lambda -1)$. We show analytically and numerically that the \textit{survivability} $S(p,l)$, which is the probability of a cluster to survive $l$ chemical shells at probability $p$, behaves near criticality as $S(p,l)=S(p_c,l)\cdot exp[(p-p_c)l/p_c]$. Thus for probabilities inside the region $|p-p_c|<p_c/l$ the behavior of the system is indistinguishable from that of the critical point.