Return times of random walk on generalized random graphs

The analytic expressions for the probability distributions of first and general return times were derived for generalized random networks. The method was useful in saving the computation time and hence obtaining return time probabilities on a much longer time scale than with simulations. It was found that the scale-free networks and ER random graphs yield slower decay of return time probabilities than the Cayley trees with the corresponding vertex degrees. The asymptotic behavior of random walk suggests that global orders are consequences in networks with heterogeneous vertex degrees such as ER random and scale-free networks.