Scaling of the Average Receiving Time on a Family of Weighted Hierarchical Networks

In this paper, based on the un-weight hierarchical networks, a family of weighted hierarchical networks are introduced, the weight factor is denoted by [Formula: see text]. The weighted hierarchical networks depend on the number of nodes in complete bipartite graph, denoted by [Formula: see text], [Formula: see text] and [Formula: see text]. Assume that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them. We deduce the analytical expression of the average receiving time (ART). The obtained remarkable results display two conditions. In the large network, when [Formula: see text], the ART grows as a power-law function of the network size [Formula: see text] with the exponent, represented by [Formula: see text], [Formula: see text]. This means that the smaller the value of [Formula: see text], the more efficient the process of receiving information. When [Formula: see text], the ART grows with increasing order [Formula: see text] as [Formula: see text] or [Formula: see text].