Horizon Visibility Graphs and Time Series Merge Trees are Dual

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.