AI 中文总结
该研究证明树-子网络结构在n^(1/3)处发生立方根相变，且gall网络的渐近计数公式仅在k=o(n^(1/3))范围内有效，与树-子网络等的最优适用范围形成对比。
AI 中文摘要
我们针对系统发育网络证明了两个令人惊讶的结果：第一，具有n个叶节点和k个网状节点的树-子网络的结构在n^(1/3)处发生急剧相变：若k=o(n^(1/3))，则随机树-子网络几乎必然是半单纯树-子网络；若k/n^(1/3)→∞且k=o(n^(1/2))，则几乎必然不是。第二，该结果表明，具有n个叶节点和固定数量k个网状节点的gall网络的渐近计数公式，在k=o(n^(1/3))范围内仍有效，超出该范围则无效。这与近期针对具有n个叶节点和k个网状节点的树-子网络及正常网络的渐近计数公式形成鲜明对比，后者在最优范围k=o(n^(1/2))内有效。
英文摘要
We prove two surprising results about phylogenetic networks. First, we show that the structure of tree-child networks with $n$ leaves and $k$ reticulation nodes undergoes a sharp phase transition at $n^{1/3}$: if $k=o(n^{1/3})$, then a random tree-child network is almost surely a semi-simplex tree-child network, whereas if $k/n^{1/3}\rightarrow\infty$ and $k=o(n^{1/2})$, it is almost surely not. Second, we show that this result implies that the asymptotic counting formula for galled networks with $n$ leaves and a fixed number $k$ of reticulation nodes remains valid in the range $k=o(n^{1/3})$, but not beyond. This is in strong contrast to recently established results for the asymptotic counting formulas for tree-child and normal networks with $n$ leaves and $k$ reticulation nodes, which are valid in the (optimal) range $k=o(n^{1/2})$.
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