  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 behaviours
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.
