Analysis of persistent and antipersistent time series with the Visibility Graph method

TL;DR

This work assesses the Reliability of the Visibility Graph (VG) method for time-series analysis across chaotic, stochastic, and colored-noise processes, linking the VG degree exponent $\gamma$ to the Hurst exponent $H$. By applying VG to ten synthetic series and estimating $\gamma$ via maximum-likelihood fits, the authors also estimate $H$ with Detrended Fluctuation Analysis and evaluate data-length effects using Kolmogorov–Smirnov tests. A key finding is that dependable $\gamma$ estimates emerge for $N \ge 1000$, with $\gamma \ge 2$ consistently when $H \le 0.5$, while $H>0.5$ yields more nuanced behavior across different processes. The study clarifies VG’s sensitivity, provides practical guidelines for data requirements, and contrasts VG with Horizontal VG (HVG) to show both correlations are detectable but not directly comparable, underscoring VG as a complementary tool for probing memory and cross-scale structure in time series.

Abstract

In this work, we investigate a range of time series, including Gaussian noises (white, pink, and blue), stochastic processes (Ornstein-Uhlenbeck, fractional Brownian motion, and Levy flights), and chaotic systems (the logistic map), using the Visibility Graph (VG) method. We focus on the minimum number of data points required to use VG and on two key descriptors: the degree distribution P(k), which often follows a power law P(k) ~ k^-gamma, and the Hurst exponent H, which identifies persistent and antipersistent time series. While the VG method has attracted growing attention in recent years, its ability to consistently characterize time series from diverse dynamical systems remains unclear. Our analysis shows that the reliable application of the VG method requires a minimum of 1000 data points. Furthermore, we find that for time series with a Hurst exponent H <= 0.5, the corresponding critical exponent satisfies gamma >= 2. These results clarify the sensitivity of the VG method and provide practical guidelines for its application in the analysis of stochastic and chaotic time series.

Figures (9)

• Figure 1: Illustration of the VG algorithm applied to the last 10 data points from a 5000-point fractional Brownian motion time series. The data points (nodes) are represented as black circles, with vertical lines indicating their heights. Purple lines depict the visibility-based connections (links) formed between the nodes.
• Figure 2: Degree distribution of the associated visibility graph created with the complete series of fractal Brownian motion of 5000 data values (for the graph, see Fig. \ref{['fig:vg_example']}), where the critical exponent of the power-law is $\gamma = 1.99 \pm 0.02$, with $k_{\text{min}} = 7$.
• Figure 3: Variation of the $\gamma$ as a function of the time series length $N$. The markers indicate the estimated $\gamma$ values for each time series, while the vertical lines represent the corresponding error bars, where in some cases (such as Ornstein-Uhlenbeck processes), the error bars are not visible due to their very small magnitude.
• Figure 4: Results for time series with $H<0.5$. Panels (a), (d), (g), and (j) show time series of 5000 data points, while panels (b), (e), (h), and (k) display the first 5000 points of time series with total length $10^5$. Panels (c), (f), (i), and (l) present the corresponding degree distributions, with black lines indicating the MLE linear fits. Specifically: (a–c) Blue Noise with $H=0.1$ and $k_{\text{min}}=6$; (d–f) fractional Brownian motion (FBM) with $H=0.2$ and $k_{\text{min}}=7$, denoted in Fig. \ref{['fig: vg_validation']} as (1); (g–i) Ornstein–Uhlenbeck process with $H=0.3$ and $k_{\text{min}}=5$, denoted in Fig. \ref{['fig: vg_validation']} as (1); and (j–l) Logistic Map with $H=0.4$ and $k_{\text{min}}=8$.
• Figure 5: Results for time series with $H=0.5$. Panels (a), (d), and (g) show time series of 5000 data points, while panels (b), (e), and (h) display the first 5000 points of the time series with total length $10^5$. Panels (c), (f), and (i) present the corresponding degree distributions, with black lines indicating the MLE linear fits. Specifically: (a–c) White Noise with $k_{\text{min}}=8$; (d–f) Ornstein–Uhlenbeck process with $k_{\text{min}}=6$, denoted in Fig. \ref{['fig: vg_validation']} as (2); and (g–i) Lévy flight with $k_{\text{min}}=8$.