  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.
