Quasiperiodic graphs: structural design, scaling and entropic properties

TL;DR

This paper introduces quasiperiodic graphs derived from time series along the quasiperiodic route to chaos, revealing their hierarchical structure, RG-based architecture, and entropy properties, advancing understanding of complex graph structures.

Contribution

It presents a novel class of quasiperiodic graphs constructed via the Horizontal Visibility algorithm, linking their structure to Farey tree hierarchy and RG theory.

Findings

Hierarchical structure inherited from Farey tree

RG theory explains graph architecture for irrational winding numbers

Graph entropy optimization recovers RG fixed-point degree distributions

Abstract

A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.