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 Erd\H{o}s-R\'enyi (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.