Trapping efficiency of random walks on weighted scale-free trees

Abstract

Trapping processes of random walks on networks have a wide range of applications and have become a research hot spot in the past few decades. Two important measures characterizing the trapping efficiency on networks are the average trapping time (ATT) and the global mean first-passage times (GMFPT). ATT is the expected time for a random walker starting from a random site to reach a fixed target site for the first time, and GMFPT is obtained from the average of ATTs over all possible target sites. In this paper, we study biased random walks on two types of weighted scale-free trees with non-fractal and fractal structures, whose weights are characterized by a real parameter w (w > 0). In the first part, we analytically evaluate the ATT on non-fractal weighted trees with a trap fixed in a hub node (node with the highest degree). Next, we present a method to calculate the GMFPT for biased random walks on general weighted networks, and analytically evaluate this quantity for fractal and non-fractal weighted scale-free trees. Finally, we analyze the effect of the parameter w on the trapping efficiency. The results for the trapping efficiency measured by the ATT for a trap fixed at a hub node show that in the non-fractal case the ATT increases monotonically as w decreases; for the trapping efficiency measured by the GMFPT, it is possible to obtain higher trapping efficiency on fractal trees by setting w properly. In contrast, for the non-fractal case, the highest trapping efficiency is produced in unweighted networks (w = 1), whereas for other values of w, the GMFPT is higher, revealing a reduction in the trapping frequency.