伪分形无标度网上平均首达时间的精确解
Exact solution for mean first-passage time on a pseudofractal scale-free web
摘要：
捕获问题的平均首次穿越时间（MFPT）的显式确定仅限于一些简单的结构，例如，规则的晶格和规则的几何分形，并确定MFPT的随机游走在其他媒体上，特别是复杂的现实网络，是一个理论上的挑战。本文研究了伪分形无标度网（PSFW）上的一个简单随机游动，其中完美陷阱位于最高度的节点上，它同时表现出真实网络中观察到的显著的无标度和小世界性质.我们得到的精确解的MFPT是通过来自PSFW的结构的递归关系计算。严格的解决方案表明，MFPT近似增加的幂律函数的节点数，与指数小于1。我们确认的封闭形式的解决方案，直接数值计算。结果表明，与规则晶格、Sierpinski分形和T图等结构相比，PSFW结构可以提高扩散输运效率。该方法可推广到其它确定性网络，使精确计算MFPT成为可能。
英文摘要：
The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially complex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.
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