Random walks between leaves of random networks

Motivated by the desire to model internet traffic we consider random walks that start and are absorbed on the leaves of random networks and study the length of such walks. We present and test two techniques to analyse these walks. On Erdős-Rényi random graphs where the probability of a walk decays exponentially with its length, the methods give indistinguishable results for the decay exponent. This simple form of decay is not apparent on heterogeneous networks such as Barabási-Albert scale free networks and in this case each technique is demonstrated to have a different strength.