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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 m and weight parameter w. 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 w , 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 w still can qualitatively, quantitatively steer leading behavior and quantitatively affect pre-factor of ATT. As to the stochastic parameter 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. [ABSTRACT FROM AUTHOR]

How to effectively control the trapping process in complex systems is of great importance in the study of trapping problem. Recently, the approach of delayed random walk has been introduced into several deterministic network models to steer trapping process. However, exploring delayed random walk on pseudo-fractal web with the co-evolution of topology and weight has remained out of reach. In this paper, we employ delayed random walk to guide trapping process on a salient deterministic weighted scale-free small-world network with the co-evolution of topology and weight. In greater detail, we first place a deep trap at one of initial nodes of the network. Then, a tunable parameter p is introduced to modulate the transition probability of random walk and dominate the trapping process. Subsequently, trapping efficiency is used as readout of trapping process and average trapping time is employed to measure trapping efficiency. Finally, the closed form solution of average trapping time (ATT) is deduced analytically, which agrees with corresponding numerical solution. The analytical solution of ATT shows that the delayed parameter p only modifies the prefactor of ATT, and keeps the leading scaling unchanged. In other words, ATT grows sublinearly with network size, whatever values p takes. In summary, the work may serves as one piece of clues for modulating trapping process toward desired efficiency on more general deterministic networks. [ABSTRACT FROM AUTHOR]