广义Vicsek分形上的精确平均首次穿越时间
Exact mean first-passage time on generalized Vicsek fractal
摘要：
分形现象广泛存在于许多复杂系统中。在本文中，我们重新审视著名的Vicsek分形，并研究它的一些结构性质的目的是了解底层拓扑结构如何影响其动力学行为。例如，我们分析确定的精确解平均首次通过时间的随机游动Vicsek分形在更轻的映射为基础的方式比以前的其他方法，包括典型的谱技术。更重要的是，我们的方法可以非常有效地精确计算基于任意允许种子生成的Vicsek分形的所有广义版本上的平均首次通过时间的解，而其他适用于典型Vicsek分形的方法将变得过于复杂，甚至失败。最后，分析结果表明，平均首达时间和顶点数之间的标度关系的广义版本的Vicsek分形保持不变的大图尺寸限制，无论是选择什么种子。
英文摘要：
Fractal phenomena may be widely observed in a great number of complex systems. In this paper, we revisit the well-known Vicsek fractal, and study some of its structural properties for purpose of understanding how the underlying topology influences its dynamic behaviors. For instance, we analytically determine the exact solution to mean first-passage time for random walks on Vicsek fractal in a more light mapping-based manner than previous other methods, including typical spectral technique. More importantly, our method can be quite efficient to precisely calculate the solutions to mean first-passage time on all generalized versions of Vicsek fractal generated based on an arbitrary allowed seed, while other previous methods suitable for typical Vicsek fractal will become prohibitively complicated and even fail. Lastly, this analytic results suggest that the scaling relation between mean first-passage time and vertex number in generalized versions of Vicsek fractal keeps unchanged in the large graph size limit no matter what seed is selected.
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