Average weighted receiving time on the non-homogeneous double-weighted fractal networks

In this paper, based on actual road networks, a model of the non-homogeneous double-weighted fractal networks is introduced depending on the number of copies s and two kinds of weight factors wi,ri(i=1,2,…,s). The double-weights represent the capacity-flowing weights and the cost-traveling weights, respectively. Denote by wijF the capacity-flowing weight connecting the nodes i and j, and denote by wijC the cost-traveling weight connecting the nodes i and j. Let wijF be related to the weight factors w1,w2,…,ws, and let wijC be related to the weight factors r1,r2,…,rs. Assuming 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. The average weighted receiving time (AWRT) is defined on the non-homogeneous double-weighted fractal networks. AWRT depends on the relationships of the number of copies s and two kinds of weight factors wi,ri(i=1,2,…,s). The obtained remarkable results display that in the large network, the AWRT grows as a power-law function of the network size Ng with the exponent, represented by θ=logs(w1r1+w2r2+⋯+wsrs)<1 when w1r1+w2r2+⋯+wsrs≠1, which means that the smaller the value of w1r1+w2r2+⋯+wsrs is, the more efficient the process of receiving information is. Especially when w1r1+w2r2+⋯+wsrs=1, AWRT grows with increasing order Ng as logNg or (logNg)2 . In the classic fractal networks, the average receiving time (ART) grows with linearly with the network size Ng. Thus, the non-homogeneous double-weighted fractal networks are more efficient than classic fractal networks in term of receiving information.