EXACT SOLUTIONS FOR AVERAGE TRAPPING TIME OF RANDOM WALKS ON WEIGHTED SCALE-FREE NETWORKS

In this paper, we present two deterministic weighted scale-free networks controlled by a weight parameter r(0 < r <= 1). One is fractal network, the other one is non-fractal network, while they have the same weight distribution when the parameter r is identical. Based on their special network structure, we study random walks on network with a trap located at a fixed node. For each network, we calculate exact solutions for average trapping time (ATT). Analyzing and comparing the obtained solutions, we find that their ATT all grow asymptotically as a powerlaw function of network order (number of nodes) with the exponent f(r) dependent on the weight parameter, but their exponent f(r) are obviously different, one is an increasing function of r, while the other is opposite. Collectively, all the obtained results show that the efficiency of trapping on weighted Scale-free networks has close relation to the weight distribution, but there is no stable positive or negative correlation between the weight distribution and the trapping time on different networks. We hope these results given in this paper could help us get deeper understanding about the weight distribution on the property and dynamics of scale-free networks.