Asymptotic enumeration of normal and hybridization networks via tree decoration
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| Format: | Preprint |
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2024
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| _version_ | 1866915202728984576 |
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| author | Fuchs, Michael Steel, Mike Zhang, Qiang |
| author_facet | Fuchs, Michael Steel, Mike Zhang, Qiang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02928 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic enumeration of normal and hybridization networks via tree decoration Fuchs, Michael Steel, Mike Zhang, Qiang Populations and Evolution 05A15, 05A16, 92D15 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. |
| title | Asymptotic enumeration of normal and hybridization networks via tree decoration |
| topic | Populations and Evolution 05A15, 05A16, 92D15 |
| url | https://arxiv.org/abs/2412.02928 |