Identifiability of local and global features of phylogenetic networks from average distances

TL;DR

This paper investigates what features of phylogenetic networks can be identified from average pairwise distances, providing theoretical insights and methods for distinguishing network structures, especially for level-1 networks.

Contribution

It characterizes the identifiable features of phylogenetic networks from pairwise distances and introduces the distance split tree as a tool for analysis.

Findings

Root and edge lengths near reticulations are not identifiable.

The distance split tree refines the network's tree of blobs.

Identifiability is limited around 4-cycles.

Abstract

Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify that the root and edge lengths adjacent to reticulations are not identifiable, and then focus on the class of zipped-up semidirected networks. We provide a criterion to swap subgraphs locally, such as 3-cycles, resulting in indistinguishable networks. We propose the "distance split tree", which can be constructed from pairwise distances, and prove that it is a refinement of the network's tree of blobs, capturing the tree-like features of the network. For level-1 networks, this distance split tree is equal to the tree of blobs refined to separate polytomies from blobs, and we prove that the mixed representation of the network is identifiable. The information loss is localized around 4-cycles, for which the placement of the reticulation is unidentifiable. The mixed representation combines split edges for 4-cycles, regular tree and hybrid edges from the semidirected network, and edge parameters that encode all information identifiable from average pairwise distances.