Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application

TL;DR

This paper derives exact topological properties of limited penetrable horizontal visibility graphs for random and chaotic time series, demonstrating their effectiveness in distinguishing chaos from randomness and analyzing real-world data.

Contribution

It provides the first exact analytical results for the degree distribution, mean degree, and clustering coefficient of these graphs, extending the visibility graph method.

Findings

Random series map to graphs with exponential degree distribution.

The method effectively discriminates chaos from randomness.

Application to real data shows practical utility.

Abstract

The limited penetrable horizontal visibility algorithm is a new time analysis tool and is a further development of the horizontal visibility algorithm. We present some exact results on the topological properties of the limited penetrable horizontal visibility graph associated with random series. We show that the random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution $P(k)\sim exp[-\lambda (k-2\rho-2)], \lambda = ln[(2\rho+3)/(2\rho+2)],\rho=0,1,2,...,k=2\rho+2,2\rho+3,...$, independent of the probability distribution from which the series was generated. We deduce the exact expressions of the mean degree and the clustering coefficient and demonstrate the long distance visibility property. Numerical simulations confirm the accuracy of our theoretical results. We then examine several deterministic chaotic series (a logistic map, the H$\acute{e}$non map, the Lorentz system, and an energy price chaotic system) and a real crude oil price series to test our results. The empirical results show that the limited penetrable horizontal visibility algorithm is direct, has a low computational cost when discriminating chaos from uncorrelated randomness, and is able to measure the global evolution characteristics of the real time series.