Scale-free properties of weighted random graphs: Minimum spanning trees and percolation

We study Erdos-Renyi random graphs with random weights associated with each link. In our approach, nodes connected by links having weights below the percolation threshold form clusters, and each cluster merges into a single node, thus generating a new "clusters network". We show that this network is scale-free with lambda = 2.5. Furthermore, we show that optimization causes the percolation threshold to emerge spontaneously, thus creating naturally a scale-free "clusters network". This phenomenon may be related to the evolution of several real world scale-free networks. Our results imply that: (i) the minimum spanning tree (MST) in random graphs is composed of percolation clusters, which are interconnected by a set of links that create a scale-free tree with lambda = 2.5 (ii) the optimal path may be partitioned into segments that follow the percolation clusters, and the lengths of these segments grow exponentially with the number of clusters that are crossed (iii) the optimal path in scale-free networks with lambda < 3 scales as l(opt) similar to log N, and the weights along the optimal path decay exponentially with their rank.