Clustering Systems of Phylogenetic Networks

TL;DR

This paper explores the relationships between various classes of phylogenetic networks and their clustering systems, providing a systematic analysis of how network types correspond to properties of their clusters.

Contribution

It establishes formal correspondences between classes of phylogenetic networks and their clustering systems, enhancing understanding of their structural relationships.

Findings

Class correspondences between network types and clustering properties

Conditions under which networks are uniquely determined by clusters

Insights into the dependencies among different network classes

Abstract

Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set $X$ of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization, that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets $\mathtt{C}(v)$ of descendant taxa of a vertex $v$. The clustering system $\mathscr{C}_N$ comprising the clusters of a network $N$ conveys key information on $N$ itself. In the special case of rooted phylogenetic trees, $T$ is uniquely determined by its clustering system $\mathscr{C}_T$. Although this is no longer true for networks in general, it is of interest to relate properties of $N$ and $\mathscr{C}_N$. Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering system of the following form: If $N$ is a network of type $\mathbb{X}$, then $\mathcal{C}_N$ satisfies $\mathbb{Y}$, and conversely if $\mathscr{C}$ is a clustering system satisfying $\mathbb{Y}$ then there is network $N$ of type $\mathbb{X}$ such that $\mathscr{C}\subseteq\mathscr{C}_N$.This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.