The Optimal Path in an Erdős-Rényi Random Graph

We study the optimal distance opt 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 Erdős-Rényi (ER) random graphs, opt scales as N, where N is the number of nodes in the graph. Thus, opt increases dramatically compared to the known small world result for the minimum distance min, which scales as logN . We also find the functional form for the probability distribution P ( opt) 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.