  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.
