Copula-based analytical results of horizontal visibility graphs for correlated time series

TL;DR

This paper derives analytical solutions for the degree distributions of horizontal visibility graphs for correlated time series using copula methods, revealing how correlations influence network properties.

Contribution

It introduces a copula-based analytical framework to understand the impact of correlations on visibility graph degree distributions, advancing theoretical understanding.

Findings

Degree distributions depend on correlation parameters.

Analytical solutions are derived up to first-order correlation effects.

Results enhance the theoretical foundation of visibility graph analysis.

Abstract

The visibility graph (VG) algorithm and its variants have been extensively studied in the time series analysis as they transform the time series into the network of nodes and links, enabling to characterize the time series in terms of network measures such as degree distributions. Despite numerous practical applications of VGs in various disciplines, analytical, rigorous understanding of VGs for the correlated time series is still far from complete due to the lack of mathematical tools for modeling the correlation structure in the time series in a tractable form. In this work, we adopt the Farlie-Gumbel-Morgenstern (FGM) copula method to derive the analytical solutions of degree distributions of the horizontal visibility graph (HVG) and its directed version (DHVG) for the correlated time series. Our analytical results show exactly how the correlation between consecutive data points affects the degree distributions of HVGs and DHVGs up to the first order of the correlation parameter in the FGM copula. Thus, our findings shed light on the rigorous understanding of the VG algorithms.