SCALING OF AVERAGE WEIGHTED RECEIVING TIME ON DOUBLE-WEIGHTED KOCH NETWORKS

Abstract

In this paper, we introduce a model of the double-weighted Koch networks based on actual road networks depending on the two weight factors w,r ∈ (0, 1]. The double weights represent the capacity-flowing weight and the cost-traveling weight, respectively. Denote by [Formula: see text] the capacity-flowing weight connecting the nodes i and j, and denote by [Formula: see text] the cost-traveling weight connecting the nodes i and j. Let [Formula: see text] be related to the weight factor w, and let [Formula: see text] be related to the weight factor r. This paper assumes that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the capacity-flowing weight of edge linking them. The weighted time for two adjacency nodes is the cost-traveling weight connecting the two nodes. We define the average weighted receiving time (AWRT) on the double-weighted Koch networks. The obtained result displays that in the large network, the AWRT grows as power-law function of the network order with the exponent, represented by θ(w,r) = ½ log2(1 + 3wr). We show that the AWRT exhibits a sublinear or linear dependence on network order. Thus, the double-weighted Koch networks are more efficient than classic Koch networks in receiving information.