On the spectral properties of Feigenbaum graphs

Abstract

A Horizontal Visibility Graph (HVG) is a simple graph extracted from an ordered sequence of real values, and this mapping has been used to provide a combinatorial encryption of time series for the task of performing network based time series analysis. While some properties of the spectrum of these graphs -such as the largest eigenvalue of the adjacency matrix-have been routinely used as measures to characterise time series complexity, a theoretic understanding of such properties is lacking. In this work we explore some algebraic and spectral properties of these graphs associated to periodic and chaotic time series. We focus on the family of Feigenbaum graphs, which are HVGs constructed in correspondence with the trajectories of one-parameter unimodal maps undergoing a period-doubling route to chaos (Feigenbaum scenario). For the set of values of the map's parameter µ for which the orbits are periodic with period 2 n , Feigenbaum graphs are fully characterised by two integers n, k and admit an algebraic structure. We explore the spectral properties of these graphs for finite n and k, and among other interesting patterns we find a scaling relation for the maximal eigenvalue and we prove some bounds explaining it. We also provide numerical and rigorous results on a few other properties including the determinant or the number of spanning trees. In a second step, we explore the set of Feigenbaum graphs obtained for the range of values of the map's parameter for which the system displays chaos. We show that in this case, Feigenbaum graphs form an ensemble for each value of µ and the system is typically weakly self-averaging. Unexpectedly, we find that while the largest eigenvalue can distinguish chaos from an iid process, it is not a good measure to quantify the chaoticity of the process, and that the eigenvalue density does a better job.

Time series analysis via network science: Concepts and algorithms

“…HVGs were used to analyze seismic signals (Telesca & Lovallo, 2012), evaluate the complex dynamics of tourism systems (Baggio & Sainaghi, 2016), construct the so‐called Feigenbaum graphs in order to study unimodal maps and their spectral properties (Flanagan, Lacasa, & Nicosia, 2019), study the volatility behavior of returns for financial time series (Zhang, Wang, & Fang, 2015), analyze heartbeat rates of healthy subjects, congestive heart failure subjects, and atrial fibrillation subjects (Xie, Han, & Zhou, 2019) and predicting catastrophes (Zhang, Xu, & Wu, 2018).…”

Horizontal visibility graph of a random restricted growth sequence

“…An illuminating characterization of HVGs using "one-point compactified" times series and tools of algebraic topology is obtained in a recent work [29]. Theoretical body of work on the HVGs includes studies of their degree distributions [14,16], information-theoretic [9,15] and other [11] topological characteristics, motifs [12,32], spectral properties [6,17], and dependence of graph features on the parameter for a specific parametric family of chaotic [4] or stochastic processes [31,34]. For more, see a recent comprehensive survey [35] and an extensive review of earlier results [23].…”

Hierarchical deposition and deterministic scale-free networks: a visibility algorithm approach

“…For the adjacency matrix spectrum, we follow the representation used in [23] and rescale the eigenvalue index for every generation n such that the smallest eigenvalue corresponds to 0 and the largest corresponds to 1. This way, we can show that the spectrum converges to a fixed shape with a hierarchical structure.…”