First-passage percolation on the random graph

We study first-passage percolation on the random graph G(p)(N) with exponen tially distributed weights on the links. For the special case of the comple te graph, this problem can be described in terms of a continuous-time Marko v chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive tr ees, which are uniform trees of N nodes, describe the structure of the clus ter once it contains all the nodes of the complete graph. From these result s, the distribution of the number of hops (links) of the shortest path betw een two arbitrary nodes is derived. We generalize this result to an asymptotic result, as N --> infinity, for t he case of the random graph where each link is present independently with a probability P-N as long as Np-N/(log N)(3) -->) infinity. The interesting point of this generalization is that (1) the limiting distribution is insen sitive to p and (2) the distribution of the number of hops of the shortest path between two arbitrary nodes has a remarkable fit with shortest path da ta measured in the Internet.