A branching process approach to level-$k$ phylogenetic networks

TL;DR

This paper introduces a novel branching process approach to analyze the structure of random level-$k$ phylogenetic networks, revealing their asymptotic tree-like shape and convergence properties.

Contribution

It is the first to apply branching process methods to study the asymptotic shape and local limits of random level-$k$ phylogenetic networks.

Findings

Networks are asymptotically tree-like in shape.

Vertex depth process converges to a Brownian excursion.

Large networks exhibit Benjamini--Schramm convergence.

Abstract

The mathematical analysis of random phylogenetic networks via analytic and algorithmic methods has received increasing attention in the past years. In the present work we introduce branching process methods to their study. This approach appears to be new in this context. Our main results focus on random level-$k$ networks with $n$ labelled leaves. Although the number of reticulation vertices in such networks is typically linear in $n$, we prove that their asymptotic global and local shape is tree-like in a well-defined sense. We show that the depth process of vertices in a large network converges towards a Brownian excursion after rescaling by $n^{-1/2}$. We also establish Benjamini--Schramm convergence of large random level-$k$ networks towards a novel random infinite network.