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 eigenvector and betweenness centralities behave as power-law decay. The scaling exponent of eigenvector centrality and the moment of eigenvalue distribution, Mn, for n ≥ 1 reveal the dependency on the Hurst exponent, H, containing the sample size effect. We also focus on persistent homology of kdimensional topological holes incorporating the filtration of simplicial complexes of associated graph. The dimension of homology group represented by Betti numbers demonstrates a strong dependency on the Hurst exponent. More precisely, the scaling exponent of the number of k-dimensional topological 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 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.