Degree distribution of the visibility graphs mapped from fractional Brownian motions and multifractal random walks

TL;DR

This paper studies the degree distributions of visibility graphs derived from fractional Brownian motions and multifractal random walks, revealing a linear relationship between the degree distribution exponent and the Hurst index, with applications to stock market data.

Contribution

It demonstrates that the degree distribution of visibility graphs from complex time series follows a power-law with an exponent linearly related to the Hurst index, highlighting the influence of temporal correlations.

Findings

Degree distributions follow power-law behavior.

Exponent $oldsymbol{ ext{α}}$ is linearly related to Hurst index $oldsymbol{ ext{H}}$.

Visibility graph degree distribution mainly depends on temporal correlation.

Abstract

The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent $\alpha$ is a linear function of the Hurst index $H$ of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from two Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the $\alpha\sim H$ linear relationship.