Persistent Homology of Weighted Visibility Graph from Fractional Gaussian Noise

Abstract

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.Comment: 17 pages, 13 figures, Comments Welcom