The trapping problem of the weighted scale-free triangulation networks for biased walks

In this paper, we study the trapping problem in the weighted scale-free triangulation networks with the growth factor m and the weight factor r. We propose two biased walks, one is standard weight-dependent walk only including the nearest-neighbor jumps, the other is mixed weight-dependent walk including both the nearest-neighbor and the next-nearest-neighbor jumps. For the weighted scale-free triangulation networks, we derive the exact analytic formulas of the average trapping time (ATT), the average of node-to-trap mean first-passage time over the whole networks, which measures the efficiency of the trapping process. The obtained results display that for two biased walks, in the large network, the ATT grows as a power-law function of the network size N-t with the exponent, represented by ln(4r+4)/ln(4m) when r not equal m - 1. Especially when the case of r = 1 and m = 2, the ATT grows linear with the network size N-t. That is the smaller the value of r, the more efficient the trapping process is. Furthermore, comparing the standard weight-dependent walk with mixed weight-dependent walk, the obtained results show that although the next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT. The smaller the value of probability parameter theta, the more efficient the trapping process for the mixed weight-dependent walk is.