Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

TL;DR

This paper analyzes the properties of horizontal visibility graphs derived from the Feigenbaum scenario, providing analytical insights into their structure, scaling, and entropy, and linking these to dynamical system characteristics.

Contribution

It offers a detailed analytical study of HV graphs in the Feigenbaum scenario, including derivations of degree distributions, scaling, and entropy, and connects these to renormalization and Lyapunov exponents.

Findings

Fixed-point graphs reveal scaling properties.

Graph entropy correlates with Lyapunov exponent.

Analytical relations for degree distributions and clustering.

Abstract

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [1] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.