Biased trapping issue on weighted hierarchical networks

In this paper, we present trapping issues of weight-dependent walks on weighted hierarchical networks which are based on the classic scale-free hierarchical networks. Assuming that edge’s weight is used as local information by a random walker, we introduce a biased walk. The biased walk is that a walker, at each step, chooses one of its neighbours with a probability proportional to the weight of the edge. We focus on a particular case with the immobile trap positioned at the hub node which has the largest degree in the weighted hierarchical networks. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping issue. Let parameter a (0 < a < 1) be the weight factor. We show that the efficiency of the trapping process depends on the parameter a; the smaller the value of a, the more efficient is the trapping process.