Asymptotic enumeration of normal and hybridization networks via tree decoration

TL;DR

This paper studies the asymptotic behavior of normal and hybridization phylogenetic networks created by random arc placements on trees, showing that most such networks are normal as the number of leaves grows, and extends enumeration methods to growing arc counts.

Contribution

The paper proves that the proportion of normal networks approaches 1 for fixed arc counts and extends asymptotic enumeration to cases where arc counts grow with the number of leaves.

Findings

Proportion of normal networks tends to 1 as leaves increase.

Asymptotic enumeration becomes straightforward for fixed arc counts.

Results extend to growing arc counts at rate o(n^{1/3}).

Abstract

Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place $k$ arcs between the edges of a rooted binary phylogenetic tree with $n$ leaves. The resulting directed graph may fail to be a phylogenetic network, and even when it is (and thereby a `tree-based' network), it may fail to be a tree-child or normal network. In this paper, we first show that if $k$ is fixed, the proportion of arc placements that result in a normal network tends to 1 as $n$ grows. From this result, the asymptotic enumeration of normal networks becomes straightforward and provides a transparent meaning to the combinatorial terms that arise. Moreover, the approach extends to allow $k$ to grow with $n$ (at the rate $o(n^\frac{1}{3})$), which was not handled in earlier work. We also investigate a subclass of normal networks of particular relevance in biology (hybridization networks) and establish that the same asymptotic results apply.