Horizon Visibility Graphs and Time Series Merge Trees are Dual

TL;DR

This paper introduces the horizon visibility graph, establishing a rigorous mathematical foundation linking it to merge trees in algebraic topology, thereby advancing nonlinear time series analysis.

Contribution

It formally defines horizon visibility graphs and proves their duality with merge trees, filling a theoretical gap in the horizontal visibility graph framework.

Findings

Horizon visibility graphs are dual to merge trees.

Horizontal visibility graphs are weak duals of merge trees.

Connections established between visibility graphs and persistent homology.

Abstract

In this paper we introduce the horizon visibility graph, a simple extension to the popular horizontal visibility graph representation of a time series, and show that it possesses a rigorous mathematical foundation in computational algebraic topology. This fills a longstanding gap in the literature on the horizontal visibility approach to nonlinear time series analysis which, despite a suite of successful applications across multiple domains, lacks a formal setting in which to prove general properties and develop natural extensions. The main finding is that horizon visibility graphs are dual to merge trees arising naturally over a filtered complex associated to a time series, while horizontal visibility graphs are weak duals of these trees. Immediate consequences include availability of tree-based reconstruction theorems, connections to results on the statistics of self-similar trees, and relations between visibility graphs and the emerging field of applied persistent homology.