论文标题
在随机均匀生长树上随机行走：平均第一学期时间的分析公式
Random walks on stochastic uniform growth trees: Analytical formula for mean first-passage time
论文作者
论文摘要
众所周知，用于确定平均第一学期时间$ \ edline {\ Mathcal {f}} $的普遍使用的方法主要基于Laplacian Spectra。但是，这种类型的方法可能会变得过于复杂，甚至在考虑到正在考虑的网络的拉普拉斯矩阵首先很难描述时也无法正常工作。在本文中，我们提出了一种有效的方法来确定一些广泛研究的树网络上的数量$ \叠加{\ Mathcal {f}} $。为此，我们首先建立了Wiener索引$ \ Mathcal {W} $和$ \ Overline {\ Mathcal {f}} $之间的通用公式。这使我们能够将问题转换为相关网络上$ \ Mathcal {W} $的计算。与以前的大多数侧重于确定性生长树的工作相反，我们的目标是考虑随机案例。为此，我们建立了一个有原则的框架，将随机性引入种植树木的过程。直接的结果，本文确定的公式对确定性案件的先前发表的结果彻底涵盖了。此外，使用建议的方法在我们的树网络上获得Kirchhoff索引也很简单。最重要的是，我们的方法比在本文考虑的情况下包括光谱技术在内的许多其他方法更容易管理。
As known, the commonly-utilized ways to determine mean first-passage time $\overline{\mathcal{F}}$ for random walk on networks are mainly based on Laplacian spectra. However, methods of this type can become prohibitively complicated and even fail to work when the Laplacian matrix of network under consideration is difficult to describe in the first place. In this paper, we propose an effective approach to determining quantity $\overline{\mathcal{F}}$ on some widely-studied tree networks. To this end, we first build up a general formula between Wiener index $\mathcal{W}$ and $\overline{\mathcal{F}}$ on a tree. This enables us to convert issues to answer into calculation of $\mathcal{W}$ on networks in question. As opposed to most of previous work focusing on deterministic growth trees, our goal is to consider stochastic case. Towards this end, we establish a principled framework where randomness is introduced into the process of growing trees. As an immediate consequence, the previously published results upon deterministic cases are thoroughly covered by formulas established in this paper. Additionally, it is also straightforward to obtain Kirchhoff index on our tree networks using the proposed approach. Most importantly, our approach is more manageable than many other methods including spectral technique in situations considered herein.
