A dissimilarity measure for semidirected networks

TL;DR

This paper introduces a new dissimilarity measure for semidirected phylogenetic networks, extending existing distances to a broader class of networks and proving its properties for tree-child networks.

Contribution

It defines a novel edge-based dissimilarity measure for semidirected networks, generalizes the Robinson-Foulds distance, and proves it is a true metric for tree-child networks.

Findings

The dissimilarity can be computed in near-quadratic time.

It extends Robinson-Foulds distance to semidirected networks.

The measure is a true distance on tree-child networks.

Abstract

Semidirected networks have received interest in evolutionary biology as the appropriate generalization of unrooted trees to networks, in which some but not all edges are directed. Yet these networks lack proper theoretical study. We define here a general class of semidirected phylogenetic networks, with a stable set of leaves, tree nodes and hybrid nodes. We prove that for these networks, if we locally choose the direction of one edge, then globally the set of directed paths starting by this edge is stable across all choices to root the network. We define an edge-based representation of semidirected phylogenetic networks and use it to define a dissimilarity between networks, which can be efficiently computed in near-quadratic time. Our dissimilarity extends the widely-used Robinson-Foulds distance on both rooted trees and unrooted trees. After generalizing the notion of tree-child networks to semidirected networks, we prove that our edge-based dissimilarity is in fact a distance on the space of tree-child semidirected phylogenetic networks.