title: Scaling of optimal-path-lengths distribution in complex networks.
abstract: We study the distribution of optimal path lengths in random graphs with random weights associated with each link ("disorder"). With each link i we associate a weight tau(i) = exp (a r(i)), where r(i) is a random number taken from a uniform distribution between 0 and 1, and the parameter a controls the strength of the disorder. We suggest, in an analogy with the average length of the optimal path, that the distribution of optimal path lengths has a universal form that is controlled by the expression (1/p(c)) (l(infinity)/a), where l(infinity) is the optimal path length in strong disorder (a --> infinity) and p(c) is the percolation threshold. This relation is supported by numerical simulations for Erdos-Rényi and scale-free graphs. We explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network.
authors.affiliation: Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel. kaliskt@mail.biu.ac.il
authors.forename: Tomer
authors.initials: T
authors.lastname: Kalisky
authors.forename: Lidia A
authors.initials: LA
authors.lastname: Braunstein
authors.forename: Sergey V
authors.initials: SV
authors.lastname: Buldyrev
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley