First passage percolation on locally treelike networks. I. Dense random graphs

We study various properties of least cost paths under independent and identically distributed (iid) disorder for the complete graph and dense Erdős–Rényi random graphs in the connected phase, with iid exponential and uniform weights on edges. Using a simple heuristic, we compute explicitly limiting distributions for (properly recentered) lengths of shortest paths between typical nodes as well as multiple source destination pairs; we also derive asymptotics for the number of edges on the shortest path, namely, the hopcount, and find that the addition of edge weights converts these graphs from ultrasmall world networks to small world networks. Finally we study the Vickrey–Clarke–Groves measure of overpayment for the complete graph with exponential edge weights and show that the complete graph is far from monopolistic for large $n$ .