Horizontal visibility graphs transformed from fractional Brownian motions: Topological properties versus the Hurst index
Abstract
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 can 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 indexes H∈(0,1). Special attention has been paid to the impact of the Hurst index on 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 nodecovering boxcounting 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 degreedegree correlations increase with 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.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 October 2011
 DOI:
 10.1016/j.physa.2011.04.020
 arXiv:
 arXiv:1012.3850
 Bibcode:
 2011PhyA..390.3592X
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Physics  Data Analysis;
 Statistics and Probability;
 Physics  Physics and Society
 EPrint:
 12 pages, 8 figures