School of Math and Information Science, Guangzhou University, Guangzhou 510006, China.
School of Cyberspace Security, Beijing University of Posts and Telecommunications, Beijing 100876, China.
J Chem Phys. 2018 Jul 14;149(2):024903. doi: 10.1063/1.5028123.
The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩ - ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v) flowers, and fractal and non-fractal scale-free trees.
首回返时间（FRT）是指随机游走者首次返回其初始位置所需的时间，而全局首次穿越时间（GFPT）则是随机游走者从随机选择的位置移动到指定位置的首次穿越时间。我们发现，在有限网络中，FRT 的方差（Var（FRT））可以表示为 Var（FRT）= 2 ⟨FRT ⟩ ⟨GFPT ⟩- ⟨FRT ⟩- ⟨FRT ⟩，其中 ⟨·⟩是随机变量的平均值。因此，我们提出了一种计算一般有限网络中 FRT 方差的方法。然后，我们使用 Var（FRT）及其变体作为指标，计算 FRT 的方差并分析规则分支网络（即 Cayley 树）上 FRT 的波动。我们发现，结果与 Sierpinski 气隙、Vicsek 分形、T 图、伪分形无标度网络、（u，v）花以及分形和非分形无标度树等其他网络中的结果不同。
