Average weighted trapping time of the node- and edge- weighted fractal networks

In this paper, we study the trapping problem in the node- and edge- weighted fractal networks with the underlying geometries, focusing on a particular case with a perfect trap located at the central node. We derive the exact analytic formulas of the average weighted trapping time (AWTT), the average of node-to-trap mean weighted first-passage time over the whole networks, in terms of the network size Ng, the number of copies s, the node-weight factor w and the edge-weight factor r. The obtained result displays that in the large network, the AWTT grows as a power-law function of the network size Ng with the exponent, represented by θ(s,r,w)=logs(srw2) when srw2 ≠ 1. Especially when srw2=1, AWTT grows with increasing order Ng as log Ng. This also means that the efficiency of the trapping process depend on three main parameters: the number of copies s > 1, node-weight factor 0 < w ≤ 1, and edge-weight factor 0 < r ≤ 1. The smaller the value of srw2 is, the more efficient the trapping process is.