Bounds on the sequence length sufficient to reconstruct binary level-$1$ phylogenetic networks under the CFN model

TL;DR

This paper establishes bounds on the genomic data length needed to accurately reconstruct binary level-1 semi-directed phylogenetic networks under the CFN model, advancing understanding of data sufficiency in network inference.

Contribution

It introduces an inference algorithm and novel quartet inference rules, showing that the required sequence length scales logarithmically or polynomially with the number of taxa.

Findings

Sequence length sufficient for accurate reconstruction scales logarithmically or polynomially with taxa.

The proposed algorithm guarantees high-probability reconstruction under the CFN model.

Novel quartet inference rules are developed for semi-directed phylogenetic networks.

Abstract

Phylogenetic trees and networks are graphs used to model evolutionary relationships, with trees representing strictly branching histories and networks allowing for events in which lineages merge, called reticulation events. While the question of data sufficiency has been studied extensively in the context of trees, it remains largely unexplored for networks. In this work we take a first step in this direction by establishing bounds on the amount of genomic data required to reconstruct binary level-$1$ semi-directed phylogenetic networks, which are binary networks in which reticulation events are indicated by directed edges, all other edges are undirected, and cycles are vertex-disjoint. For this class, methods have been developed recently that are statistically consistent. Roughly speaking, such methods are guaranteed to reconstruct the correct network assuming infinitely long genomic sequences. Here we consider the question whether networks from this class can be uniquely and correctly reconstructed from finite sequences. Specifically, we present an inference algorithm that takes as input genetic sequence data, and demonstrate that the sequence length sufficient to reconstruct the correct network with high probability, under the CFN model of evolution, scales logarithmically, polynomially, or polylogarithmically with the number of taxa, depending on the parameter regime. As part of our contribution, we also present novel inference rules for quartet data in the semi-directed phylogenetic network setting.