Scaling of average sending time on weighted Koch networks

Abstract

Random walks on weighted complex networks, especially scale-free networks, have attracted considerable interest in the past. But the efficiency of a hub sending information on scale-free small-world networks has been addressed less. In this paper, we study random walks on a class of weighted Koch networks with scaling factor 0 < r ⩽ 1. We derive some basic properties for random walks on the weighted Koch networks, based on which we calculate analytically the average sending time (AST) defined as the average of mean first-passage times (MFPTs) from a hub node to all other nodes, excluding the hub itself. The obtained result displays that for 0 < r < 1 in large networks the AST grows as a power-law function of the network order with the exponent, represented by \documentclass[12pt]{minimal}\begin{document}$\log _{4}\frac{3r+1}{r}$\end{document}log43r+1r, and for r = 1 in large networks the AST grows with network order as N ln N, which is larger than the linear scaling of the average receiving time defined as the average of MFPTs for random walks to a given hub node averaged over all starting points.