LAUSR

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 n 2 = 4 n 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 | N k | .