Asymptotic Enumeration and Distributional Properties of Galled Networks

TL;DR

This paper provides the first asymptotic enumeration and distributional analysis of galled networks with a large number of leaves, revealing their growth rate and reticulation node distribution.

Contribution

It introduces the first asymptotic counting result for large galled networks and determines the limiting distribution of reticulation nodes.

Findings

Asymptotic count of galled networks with n leaves

Limiting distribution of reticulation nodes in random galled networks

Application of advanced asymptotic analysis techniques

Abstract

We show a first-order asymptotics result for the number of galled networks with $n$ leaves. This is the first class of phylogenetic networks of {\it large} size for which an asymptotic counting result of such strength can be obtained. In addition, we also find the limiting distribution of the number of reticulation nodes of a galled networks with $n$ leaves chosen uniformly at random. These results are obtained by performing an asymptotic analysis of a recent approach of Gunawan, Rathin, and Zhang (2020) which was devised for the purpose of (exactly) counting galled networks. Moreover, an old result of Bender and Richmond (1984) plays a crucial role in our proofs, too.