Unveiling the impact of weight heterogeneity on random walks in weighted extended tree-like fractals

Abstract

In this paper, we put forward a class of weighted extended tree-like fractals and further use them as test bed to unveil the impact of weight heterogeneity on random walks. Specifically, a family of weighted extended tree-like fractals are first proposed, which are parameterized by a growth parameter [Formula: see text] and weight parameter [Formula: see text]. Then, we explore standard weight-dependent walk on the networks by deploying three traps at initial three nodes. To this end, we derive analytically the average trapping time (ATT) to measure the trapping efficiency and the obtained results show that depending on values of [Formula: see text], ATT may grow sub-linearly, linearly and super-linearly with the network size. Besides, it can also quantitatively impact the leading behavior and pre-factor of ATT simultaneously. Finally, more challenging mixed weight-dependent random walk that takes non-nearest-neighbor hopping is addressed. Analytical solutions of ATT derived under this new scenario imply that weight parameter [Formula: see text] still can qualitatively, quantitatively steer leading behavior and quantitatively affect pre-factor of ATT. As to the stochastic parameter [Formula: see text] controlling mixed random walk, it could only impact the pre-factor of ATT and only have negligible effect on the leading behavior of ATT. In summary, this work could further augment our understanding of random walks on networks.