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 Delta p(c) for systems of finite size. Here we present evidence that for complex networks Delta p(c)similar to p(c)/center dot, where center dot similar to N-opt(nu) is the average length of the percolation cluster, and N is the number of nodes in the network. For Erdos-Renyi graphs nu(opt)=1/3, while for scale-free networks with a degree distribution P(k)similar to k(-lambda) and 3 <lambda < 4, nu(opt)=(lambda-3)/(lambda-1). We show analytically and numerically that the survivability S(p,center dot), which is the probability of a cluster to survive center dot chemical shells at probability p, behaves near criticality as S(p,center dot)=S(p(c),center dot)exp[(p-p(c))center dot/p(c)]. Thus for probabilities inside the region parallel to p-p(c)parallel to < p(c)/center dot the behavior of the system is indistinguishable from that of the critical point.