Summary: 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 \(0 < r \leq 1\). 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 \(\log_{4}\frac{3r+1}{r}\), 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.