In this paper, we utilize persistent homology technique to examine the topological properties of the visibility graph constructed from fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm and the standard network in addition to the global properties in the context of topology, will be examined. Our results demonstrate that the distribution of {\it eigenvector} and {\it betweenness centralities} behave as power-law decay. The scaling exponent of {\it eigenvector centrality} and the moment of {\it eigenvalue} distribution, $M_{n}$, for $n\ge1$ reveal the dependency on the Hurst exponent, $H$, containing the sample size effect. We also focus on persistent homology of $k$-dimensional topological holes incorporating the filtration of simplicial complexes of associated graph. The dimension of homology group represented by {\it Betti numbers} demonstrates a strong dependency on the Hurst exponent. More precisely, the scaling exponent of the number of $k$-dimensional topological \textit{holes} appearing and disappearing at a given threshold, depends on $H$ which is almost not affected by finite sample size. We show that the distribution function of \textit{lifetime} for $k$-dimensional topological holes decay exponentially and corresponding slope is an increasing function versus $H$ and more interestingly, the sample size effect is completely disappeared in this quantity. The persistence entropy logarithmically grows with the size of visibility graph of system with almost $H$-dependent prefactors.

中文翻译：

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基于分数阶高斯噪声的加权可见性图的持久同伦性

在本文中，我们使用持久性同源技术检查由分数高斯噪声（fGn）构成的可见性图的拓扑特性。我们开发了加权自然可见性图算法，除了拓扑结构中的全局属性外，还将检查标准网络。我们的结果证明{\ it特征向量}和{\ it中间性中心}的分布表现为幂律衰减。{\ it特征向量中心性}的缩放指数和{\ it特征值}分布的时刻$ M_ {n} $，对于$ n \ ge1 $，揭示了对包含样本大小的Hurst指数$ H $的依赖性。影响。我们还将重点放在$ k $维拓扑孔的持久同源性中，并结合了相关图的简单复杂性的过滤。由{\ it贝蒂数}表示的同源基团的维数显示出对Hurst指数的强烈依赖性。更精确地，在给定阈值处出现和消失的$ k $维拓扑\ textit {holes}的数量的缩放指数取决于$ H $，几乎不受有限样本大小的影响。我们显示，\ textit {lifetime}在$ k $维拓扑孔中的分布函数呈指数衰减，并且对应的斜率是对$ H $的递增函数，更有趣的是，样本数量效应在此数量上完全消失了。持久性熵随具有几乎$ H $依赖因素的系统的可见性图的大小呈对数增长。在给定的阈值处出现和消失的$ k $维拓扑\ textit {holes}数量的缩放指数取决于$ H $，而H $几乎不受有限样本量的影响。我们显示，\ textit {lifetime}在$ k $维拓扑孔中的分布函数呈指数衰减，并且对应的斜率是对$ H $的递增函数，更有趣的是，样本数量效应在此数量上完全消失了。持久性熵随具有几乎$ H $依赖因素的系统的可见性图的大小呈对数增长。在给定的阈值处出现和消失的$ k $维拓扑\ textit {holes}数量的缩放指数取决于$ H $，而H $几乎不受有限样本量的影响。我们显示，\ textit {lifetime}在$ k $维拓扑孔中的分布函数呈指数衰减，并且对应的斜率是对$ H $的递增函数，更有趣的是，样本数量效应在此数量上完全消失了。持久性熵随具有几乎$ H $依赖因素的系统的可见性图的大小呈对数增长。我们显示，\ textit {lifetime}在$ k $维拓扑孔中的分布函数呈指数衰减，并且对应的斜率是对$ H $的递增函数，更有趣的是，样本数量效应在此数量上完全消失了。持久性熵随具有几乎$ H $依赖因素的系统的可见性图的大小呈对数增长。我们显示，\ textit {lifetime}在$ k $维拓扑孔中的分布函数呈指数衰减，并且对应的斜率是对$ H $的递增函数，更有趣的是，样本数量效应在此数量上完全消失了。持久性熵随具有几乎$ H $依赖因素的系统的可见性图的大小呈对数增长。