The Optimal Pathin an Erdős-Rényi Random Graph

Abstract

We study the optimal distance in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path (cost) is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in Erds-R{é}nyi (ER) random graphs, scales as N1/3, where N is the number of nodes in the graph. Thus, increases dramatically compared to the known small world result for the minimum distance , which scales as logN. We also find the functional form for the probability distribution of optimal paths. In addition we show how the problem of strong disorder on a random graph can be mapped onto a percolation problem on a Cayley tree and using this mapping, obtain the probability distribution of the maximal weight on the optimal path.