The space of equidistant phylogenetic cactuses

We introduce and investigate the space of equidistant X-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 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.