How tree-based is my network? Proximity measures for unrooted phylogenetic networks

Abstract

Evolutionary events like hybridization and horizontal gene transfer cannot be represented by a tree but require a phylogenetic network instead. Nonetheless, it has been argued that evolution is fundamentally tree-like, in the sense that it can be modeled by a tree with additional arcs. Phylogenetic networks with this property are called tree-based, and this concept has recently attracted significant attention in evolutionary research. But classifying networks into treebased and non-tree-based ones is usually not sufficient for biological purposes, because sometimes tree-basedness is missed merely due to incomplete sampling. Therefore, measures are needed to quantify how close a non-tree-based network is to being tree-based. Such measures naturally also lead to characterizations of tree-based networks, and some measures for this purpose have recently been introduced both for binary and non-binary rooted phylogenetic networks. In the present manuscript, we generalize all these measures to unrooted networks, for which tree-basedness is mathematically more involved, and also present some new measures. This leads to new characterizations of unrooted tree-based networks. We analyze the relationships of the new proximity measures to one another to demonstrate their common properties as well as their respective differences.

A Structure Theorem for Rooted Binary Phylogenetic Networks and Its Implications for Tree-Based Networks

“…For example, given an unrooted binary phylogenetic network N , the problem of deciding whether or not N is tree-based is NP-complete (Theorem 2 in [6]). This implies that, as Fischer and Francis [4] Fig. 9.…”

“…Related to the decision problem, discussion has also been made on what we call the deviation quantification problem that requires computing the deviation measure \delta (N ) of a phylogenetic network N from being tree-based [4,8,18,22,23]. More precisely, based on the point that any non-tree-based rooted binary phylogenetic network N can be converted into a tree-based network by introducing new leaves [9], several studies Downloaded 11/03/21 to 44.224.250.200 Redistribution subject to CCBY license suggested measuring the degree of the deviation of N from being tree-based by the minimum number \delta (N ) of leaves that need to be attached to make N tree-based [8,18] (or some alternative indices equivalent to \delta (N ) [8]).…”

“…Structure of unrooted phylogenetic networks. The notion of unrooted (undirected) tree-based phylogenetic networks was originally defined in [6] and has been discussed in several studies in recent years (e.g., [4,5,14]). However, in contrast to the rooted case, there are many computational difficulties in this direction even in binary settings.…”

“…measuring deviations from being tree-based proposed in [4] are NP-hard to compute. Considering that phylogenetic networks reconstructed using distance data are necessarily unrooted, it would be important to explore subclasses of unrooted phylogenetic networks with a mathematically nice structure in order to develop new and useful methods for analyzing biological data.…”

“…The above thoughts are closely related to the motivation behind the concepts of ``tree-based phylogenetic networks"" and their underlying ``subdivision trees"" (i.e., phylogenetic trees embedded in tree-based phylogenetic networks in a certain form), which were originally introduced by Francis and Steel in [9] and have been extensively studied in the last few years (e.g., [1,2,4,6,7,8,11,14,18,19,23,25,29]). Treebased phylogenetic networks are a biologically meaningful extension of phylogenetic trees and form a fairly large class of networks that encompasses many popular subclasses of phylogenetic networks, such as tree-child networks, tree-sibling networks, and reticulation-visible networks [9].…”