The modified box dimension and average weighted receiving time of the weighted hierarchical graph

Abstract

In this paper we study the weighted hierarchical graph which is generated from bipartite graph with N=n1+n2 vertices, in which the weights of edges have been assigned to different values with certain scale. Firstly, we introduce the definition of the modified box dimension. Then for the weighted hierarchical graph we deduce the modified box dimension, dimMB({Gn}n∈N)=−logrN, depending on the weighted factor r and the number N of copies. Secondly, we mainly study their two average weighted receiving times (AWRTs), 〈T〉In and 〈T〉IIn, of the weighted hierarchical graph on random walk. We discuss two cases. In the case of n1n2r≠n2−n1, we deduce both AWRTs grow as a power-law function of the network size |V(Gn)| with the postive exponent, represented by θ=logN(Nn1n2) or θ=logNr=1−dimMB({Gn}n∈N), which means that the bigger the value of the modified box dimension is, the slower the process of receiving information is. In the case of n1n2r=n2−n1, both AWRTs tend to constant ( if N<n1n2), the AWRTs grow with increasing order as logN|V(Gn)|( if N=n1n2), and both AWRTs grow as a power-law function of the network size |V(Gn)| with the exponent, represented by θ=logN(Nn1n2)>0( if N>n1n2).