Effects of node position on diffusion and trapping efficiency for random walks on fractal scale-free trees

Abstract

We study unbiased discrete random walks on the FSFT based on the its self-similar structure and the relations between random walks and electrical networks. First, we provide new methods to derive analytic solutions of the MFPT for any pair of nodes, the MTT for any target node and MDT for any starting node. And then, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we analyze the effect of trap's position on trapping efficiency and the effect of starting position on diffusion efficiency. Comparing the trapping efficiency and diffusion efficiency among all nodes of FSFT, we find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that: the node which is at the center of FSFT is the best trapping site, but it is also the worst diffusing site. The nodes which are the farthest nodes from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maximum and minimum of MTT and MDT, we found that the maximum of MTT is almost $\frac{20m^2+32m+12}{4m^2+4m+1}$ times of the minimum of MTT, but the the maximum of MDT is almost equal to the minimum of MDT. These results shows that the position of target node has big effect on trapping efficiency, but the position of starting node almost has no effect on diffusion efficiency. We also conducted numerical simulation to test the results we have derived, the results we derived are consistent with those obtained by numerical simulation.