Width of percolation transition in complex networks

It is known that the critical probability for the percolation transition is not a sharp threshold. Actually it is a region of nonzero width Δ pc for systems of finite size. Here we present evidence that for complex networks Δ pc ∼ pc/ℓ., where ℓ∼ N νopt is the average length of the percolation cluster, and N is the number of nodes in the network. For Erdos-Rényi graphs νopt =1/3, while for scale-free networks with a degree distribution P (k) ∼ k-λ and 3<λ<4, νopt = (λ-3) (λ-1). We show analytically and numerically that the 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 (pc, ℓ) exp [(p- pc) ℓ/pc]. Thus for probabilities inside the region |p- pc | < pc ℓ the behavior of the system is indistinguishable from that of the critical point.