Spaces of ranked tree-child networks

TL;DR

The paper develops a dual framework for comparing binary ranked tree-child networks (RTCNs): a discrete, poset-guided encoding that yields a generalized rNNI-style distance on RTCNs, and a continuous, CAT(0)-orthant space of equidistant tree-child networks (ETCNs) that enables efficient geodesic computations. Central to the approach is encoding RTCNs as maximal chains of cluster systems on leaf set $X$, via the poset $(\mathfrak{T}(X),\preceq)$, which unifies trees and networks and extends to non-binary cases. The authors prove bijection between binary RTCNs and maximal chains, introduce rNNI$^*$ moves to connect RTCNs, and construct the CAT(0)-orthant space $\mathfrak{S}(X)$ of ETCNs with a polynomial-time distance $\delta$. These results generalize known spaces for ultrametric trees, provide a principled method to compare RTCNs, and open avenues for statistical analysis on the space of tree-child 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.

Figures (10)

• Figure 1: (a) A binary RTCN on the set $X = \{a,b,c,d,e\}$. Each dotted horizontal line corresponds to vertices that have the same rank. (b) An ETCN on $X$ obtained by assigning suitable weights to the arcs of the RTCN in (a).
• Figure 2: An example of the process that generates a binary RTCN on $X=\{a,b,c,d\}$. (a) The result of Step 1. (b) The result of performing (1) in Step 2. (c) The result of performing (2) in Step 3. (d) The resulting binary RTCN after Step $n=4$. Vertices of rank $i$ are drawn on the dotted horizontal line numbered $i$$(1 \leq i \leq 4)$.
• Figure 3: The two operations (a) $\vdash_{(1)}$ and (b) $\vdash_{(2)}$ that can be applied to a cluster system and how they are related to the process of generating binary RTCNs.
• Figure 4: The two types of modifications on binary ranked trees allowed in an rNNI: (a) Swapping the ranks of vertices $u$ and $v$. (b) An actual nearest neighbor interchange.
• Figure 5: Two consecutive rNNI$^*$s that transform the binary RTCN $\mathcal{N}$ on $X=\{a,b,c,d,e,f\}$ through the intermediate binary RTCN $\mathcal{N}'$ to the binary ranked tree $\mathcal{T}$.

Theorems & Definitions (6)

• Lemma 3.1
• Theorem 3.2
• Theorem 4.2
• Example 5.1
• Theorem 5.2
• Theorem 6.1