Random walks in small-world exponential treelike networks

In this paper, we investigate random walks in a family of small-world trees having an exponential degree distribution. First, we address a trapping problem, that is, a particular case of random walks with an immobile trap located at the initial node. We obtain the exact mean trapping time defined as the average of the first-passage times (FPT) from all nodes to the trap, which scales linearly with the network order N in large networks. Then, we determine analytically the mean sending time, which is the mean of the FPTs from the initial node to all other nodes, and show that it grows with N, varying approximately as N lnN. After that, we compute the precise global mean first-passage time among all pairs of nodes and find that it also varies approximately as N lnN in the large limit of N. After obtaining the relevant quantities, we compare them with each other and relate our results to the efficiency for information transmission by regarding the walker as an information messenger. Finally, we compare our results with those previously reported for other trees with different structural properties (e. g., degree distribution), such as the standard fractal trees and the scale-free small-world trees, and show that the shortest path between a pair of nodes in a tree is responsible for the scaling of the FPT between the two nodes.