  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.
