The average trapping time of non-nearest-neighbor jumps on nested networks

Abstract

In this paper, we consider the trapping problem on the nearest-neighbor (NN) and non-nearest-neighbor (NNN) jumps on nested networks. Based on the nested construction of the network and the use of probability generating function tool, the iterative rules of two successive generations of the network are found, and the analytical expression of the average trapping time (ATT) is finally obtained. We allow two jump modes in the network at the same time, and the results show that the choice probability of the jump mode is not related to the exponential term of the scaling expression, but to its leading factor term. According to the analytic solution of ATT, we can find that the value of ATT expands superlinearly with the increase of network size. In addition, the numerical simulation results of parameters q (the probability of choosing NNN jump) and n (the generation of the network) show that with fixed n, ATT decreases with the increase of q; while with fixed q, ATT increases with the increase of n. In summary, this work can observe the effect of different hopping modes on random walk efficiency in complex networks.