Scaling of the average weighted receiving time on a family of double-weighted hierarchical networks

Abstract

In this paper, we introduce a family of double-weighted hierarchical networks, which is depended on the initial complete bipartite graph and two weight factors r, w (0<r<1, 0<w<1). Then we consider the biased walk in the double-weighted hierarchical networks. What is more, we define two weighted times, the mean weighted first-passing time (MWFPT) and the average weighted receiving time (AWRT). According to the definition, we calculate the exact expressions of the MWFPT and the AWRT on the double-weighted hierarchical networks. The expressions we obtain in this paper illustrate that the AWRT tends to a constant when the size of the networks increases, while if n2=4n1n2(1−w2) (where n1 and n2 denote the number of nodes in initial complete bipartite graph , and n=n1+n2) holds, the AWRT grows as a power-law function of network order with the exponent, denoted by logn|Nk|.