Optimal networks revealed by global mean first return time

Abstract

Random walks have wide application in real lives, such as target search, reaction kinetics, polymer chains, and so on. In this paper, we consider discrete random walks on general connected networks and focus on the global mean first return time (GMFRT), which is defined as the mean first return time averaged over all the possible starting positions (vertices), aiming at finding the structures which have the maximal (or the minimal) GMFRT. Our results show that, among all trees with a given number of vertices, trees with linear structure are those with the minimal GMFRT and stars are those with the maximal GMFRT. We also find that, among all unweighted and undirected connected simple graphs with a given number of edges and vertices, the graphs maximizing (resp. minimizing) the GMFRT are the ones for which the variance of the nodes degrees is the largest (resp. the smallest).