Scaling of average weighted shortest path and average receiving time on weighted hierarchical networks

Recent work on the networks has focused on the weighted hierarchical networks that are significantly different from the un-weighted hierarchical networks. In this paper we study a family of weighted hierarchical networks which are recursively defined from an initial uncompleted graph, in which the weights of edges have been assigned to different values with certain scale. Firstly, we study analytically the average weighted shortest path (AWSP) on the weighted hierarchical networks. Using a recursive method, we determine explicitly the AWSP. The obtained rigorous solution shows that the networks grow unbounded but with the logarithm of the network size, while the weighted shortest paths stay bounded. Then, depending on a biased random walk, we research the mean first-passage time (MFPT) between a hub node and any peripheral node. Finally, we deduce the analytical expression of the average of MFPTs for a random walker originating from any node to first visit to a hub node, which is named as the average receiving time (ART). The obtained result shows that ART is bounded or grows sublinearly with the network order relating to the number of initial nodes and the weighted factor or grows quadratically with the iteration.