The space of equidistant phylogenetic cactuses

TL;DR

This paper introduces the space of equidistant phylogenetic cactuses, proves it is a CAT(0) space, and explores its combinatorial and geometric properties to facilitate statistical analysis of phylogenetic networks.

Contribution

It defines the space of equidistant cactuses, proves it is CAT(0), and provides combinatorial encodings linking networks to set-theoretic structures, extending phylogenetic analysis tools.

Findings

Equidistant-cactus space is a CAT(0) metric space.

Ranked rooted X-cactuses can be encoded via set-theoretic conditions.

The space includes ultrametric trees as a subset.

Abstract

We introduce and investigate the space of \emph{equidistant} $X$-\emph{cactuses}. These are rooted, arc weighted, phylogenetic networks with leaf set $X$, where $X$ is a finite set of species, and all leaves have the same distance from the root. The space contains as a subset the space of ultrametric trees on $X$ that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning \emph{ranked} rooted $X$-cactuses. In particular, we show that such networks can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of $X$ that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted $X$-trees in terms of partitions of $X$, which provides an alternative proof that the space of ultrametric trees on $X$ is CAT(0). As with spaces of phylogenetic trees, we expect that our results should provide the basis for and new directions in performing statistical analyses for collections of phylogenetic networks with arc lengths.