Width of percolation transition in complex networks

TL;DR

This paper investigates the width of the percolation transition in complex networks, revealing how it scales with network size and cluster length, and providing analytical and numerical insights into cluster survivability near criticality.

Contribution

It introduces a scaling relation for the transition width in complex networks and analyzes the survivability function near criticality.

Findings

Transition width scales as Δp_c ~ p_c / l

Survivability S(p,l) behaves as S(p_c,l) * exp[(p-p_c)l/p_c]

Behavior near criticality is indistinguishable within |p-p_c| < p_c/l

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.