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

We study unbiased discrete random walks on fractal scale-free trees (FSFTs) based on their self-similar structure and the relations between random walks and electrical networks. Firstly, we provide new methods to derive analytic solutions of the mean first-passage time (MFPT) for any pair of nodes, the mean trapping time (MTT) for any target node and the mean diffusing time (MDT) for any starting node. Then, using the MTT and the MDT as the measures of trapping efficiency and diffusion efficiency respectively, we analyze the effect of a trap's position on the trapping efficiency and the effect of the starting position on the diffusion efficiency. Comparing the trapping efficiency and diffusion efficiency among all nodes of an FSFT, we find the best (or worst) trapping sites and the best (or worst) diffusing sites. Our results show that the node at the center of the FSFT is the best trapping site, but it is also the worst diffusing site. The nodes that are farthest from the two hubs are the worst trapping sites, but they are also the best diffusion sites. Comparing the maxima of the MTT and MDT with their minima, we find that the maximum of the MTT is about (20m² + 32m + 12)/(4m² + 4m + 1) times the minimum of the MTT, whereas the maximum of the MDT is almost equal to the minimum of the MDT. These results show that the position of the target node has a large effect on the trapping efficiency, but the position of the starting node has almost no effect on the diffusion efficiency. We also conducted numerical simulations which showed they are in good agreement with the derived results.