Random Walks between Leaves of Random Networks

We consider random walks that start and are absorbed on the leaves of random networks and study the length of such walks. For the networks we investigate, Erdos-Renyi random graphs and Barabasi-Albert scale free networks, these walks are not transient and we consider various approaches to computing the probability of a given length walk.One approach is to label nodes according to both their total degree and the number of links connected to leaf nodes, and as a byproduct we compute the probability of a random node of a scale free network having such a label.