Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process c
an be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst index $Hin(0,1)$. Special attention has been paid to the impact of Hurst index on the topological properties. It is found that the clustering coefficient $C$ decreases when $H$ increases. We also found that the mean length $L$ of the shortest paths increases exponentially with $H$ for fixed length $N$ of the original time series. In addition, $L$ increases linearly with respect to $N$ when $H$ is close to 1 and in a logarithmic form when $H$ is close to 0. Although the occurrence of different motifs changes with $H$, the motif rank pattern remains unchanged for different $H$. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension $d_B$ decreases with $H$. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with the degree, which indicate that the horizontal visibility graphs are assortative. With the increase of $H$, the Pearson coefficient decreases first and then increases, in which the turning point is around $H=0.6$. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative.
