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 Deltap(c) for systems of finite size. Here we present evidence that for complex networks Deltap(c) approximately p(c)/l, where l approximately Nnu(opt), where is the average length of the percolation cluster, and N is the number of nodes in the network. For Erdos-Rényi graphs nu(opt)=1/3, while for scale-free networks with a degree distribution P(k) approximately k(-lambda) and 3<lambda<4, nu(opt)=(lambda-3)/(lambda-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(p(c),l)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.
Kalisky
Tomer
T
Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel. kaliskt@mail.biu.ac.il
Cohen
Reuven
R