First passage percolation on the random graph

We study first passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph this problem can be described in terms of a continuous time Markov 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 trees, which are uniform trees of N nodes, describe the structure of the cluster once it contains all the nodes of the complete graph. From these results the distribution of the number of hops (links) of the shortest path between two arbitrary nodes is derived. We generalize this result to an asymptotic result, as N → ∞, for the case of the random graph where each link is present independently with a probability pN as long as NpN (logN)3 →∞. The interesting point of this generalization is that (i) the limiting distribution is insensitive to p and (ii) the distribution of the number of hops of the shortest path between two arbitrary nodes has a remarkable fit with shortest path data measured in the Internet (cf. [6]).