Abstract
First-passage processes on fractals are of particular importance since fractals are ubiquitous in nature, and first-passage processes are fundamental dynamic processes that have wide applications. The global mean first-passage time (GMFPT), which is the expected time for a walker (or a particle) to first reach the given target site while the probability distribution for the position of target site is uniform, is a useful indicator for the transport efficiency of the whole network. The smaller the GMFPT, the faster the mass is transported on the network. In this work, we consider the first-passage process on a class of fractal scale-free trees (FSTs), aiming at speeding up the first-passage process on the FSTs. Firstly, we analyze the global mean first-passage time (GMFPT) for unbiased random walks on the FSTs. Then we introduce proper weight, dominated by a parameter w (w > 0), to each edge of the FSTs and construct a biased random walks strategy based on these weights. Next, we analytically evaluated the GMFPT for biased random walks on the FSTs. The exact results of the GMFPT for unbiased and biased random walks on the FSTs are both obtained. Finally, we view the GMFPT as a function of parameter w and find the point where the GMFPT achieves its minimum. The exact result is obtained and a way to optimize and speed up the first-passage process on the FSTs is presented.
Highlights
Some of them focus on disclosing the effects of the topology on the mean first-passage time (MFPT), and lots of results have been obtained for unbiased random walks on different networks, such as Sierpinski gaskets [19,20], pseudofractal scale-free web [21,22], scale-free Koch networks [9,23], (u, v) flowers [24], and many fractal scale-free trees [9,21,25,26,27]
For any n ≥ 0, the global mean first-passage time (GMFPT) for biased random walks on the weighted fractal scale-free trees G (n) can be expressed as [36]
Recalling the exact result of the GMFPT, as shown in Equation (22), for a biased random walk on the weighted networks, we find for a network that is big enough, i.e, n → ∞, Equation (22) can be rewritten as
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Some of them focus on disclosing the effects of the topology on the MFPT (or GMFPT), and lots of results have been obtained for unbiased random walks on different networks, such as Sierpinski gaskets [19,20], pseudofractal scale-free web [21,22], scale-free Koch networks [9,23], (u, v) flowers [24], and many fractal scale-free trees [9,21,25,26,27]. By introducing the proper weight to each edge of the network and designing a proper biased random walk strategy, one can shorten the MFPT to obtain higher transport efficiency on the underling networks [31,32,33,34,35]. We study unbiased and biased random walks on the general fractal scale-free trees (FSTs), aiming at shortening the GMFPT and optimizing the transport efficiency of the networks.
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