Spaces of ranked tree-child networks

TL;DR

This paper introduces new methods to compare ranked tree-child networks by encoding them with partially ordered sets, defining two new spaces, and establishing properties like CAT(0)-space for efficient distance computation.

Contribution

It presents a novel encoding of ranked tree-child networks and defines two new metric spaces, extending existing phylogenetic network comparison frameworks.

Findings

Defined two new spaces of ranked binary tree-child networks.

Proved the continuous space is a CAT(0)-orthant space.

Enabled efficient computation of distances between equidistant networks.

Abstract

Ranked tree-child networks are a recently introduced class of rooted phylogenetic networks in which the evolutionary events represented by the network are ordered so as to respect the flow of time. This class includes the well-studied ranked phylogenetic trees (also known as ranked genealogies). An important problem in phylogenetic analysis is to define distances between phylogenetic trees and networks in order to systematically compare them. Various distances have been defined on ranked binary phylogenetic trees, but very little is known about comparing ranked tree-child networks. In this paper, we introduce an approach to compare binary ranked tree-child networks on the same leaf set that is based on a new encoding of such networks that is given in terms of a certain partially ordered set. This allows us to define two new spaces of ranked binary tree-child networks. The first space can be considered as a generalization of the recently introduced space of ranked binary phylogenetic trees whose distance is defined in terms of ranked nearest neighbor interchange moves. The second space is a continuous space that captures all equidistant tree-child networks and generalizes the space of ultrametric trees. In particular, we show that this continuous space is a so-called CAT(0)-orthant space which, for example, implies that the distance between two equidistant tree-child networks can be efficiently computed.