Approximation for return time distributions of random walks on sparse networks

We propose an approximation for the first return time distribution of random walks on undirected networks. We combine a message-passing solution with a mean-field approximation to account for the short- and long-term behaviors, respectively. We test this approximation on several classes of large graphs and find excellent agreement between our approximations and the true distributions. While the statistical properties of a random walk will depend on the structure of the network, the observed agreement between our approximations and numerical calculations implies that while local structure is clearly very influential, global structure is only important in a relatively superficial way, namely through the total number of edges.