LAZY RANDOM WALKS ON PSEUDOFRACTAL SCALE-FREE WEB WITH A PERFECT TRAP

Abstract

Lazy random walks have been used in many scientific fields such as image segmentation and optimal transport; however, related theoretical results are much less for this dynamical process. In this paper, we study lazy random walks in a pseudofractal scale-free web, where the self-loop jumps on graph vertexes are considered. For a special random walks with one trap fixed at a hub node, also known as the trapping problem, we derive the exact analytic formulas of the average trapping time (ATT), an indicator measuring the efficiency of the trapping process, by using two different methods. The results obtained by the two methods are consistent. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule with the self-loop jumps. Specifically, adding the self-loop to change the walking rule can affect the coefficient of the ATT formula, but it cannot change the leading scaling of the trapping efficiency. We hope that these results in this paper can help us better understand the biased random walk process in complex systems.