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

摘要

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 n(2) = 4n(1) n(2)(1 - w(2)) (where n(1) and n(2) denote the number of nodes in initial complete bipartite graph and n = n(1) + n(2)) holds, the AWRT grows as a power-law function of network order with the exponent, denoted by log(n) vertical bar N-k vertical bar. (C) 2019 Elsevier B.V. All rights reserved.