通过自同构权重的无限制二元系统发育网络的精确与渐近计数
Exact and asymptotic enumeration of unrestricted binary phylogenetic networks through automorphism weights
- CRISP – Centre de Recerca Independent de sa Pobla（萨波布拉独立研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过自同构权重对无限制二元系统发育网络进行精确与渐近计数，提出递归方法，给出k≤5的闭式计数，并证明非树-子网络占比及随机网络非平凡自同构概率的渐近结果。
AI中文摘要：
设 $\cP_{\ell,k}$ 为具有 $\ell$ 个标记叶子、$k$ 个网状化事件且无平行边的有根二元系统发育网络的集合。我们记 $|\cP_{\ell,k}|=W_k(\ell)+D_k(\ell)$，其中加权计数 $W_k(\ell)$ 加上叶子固定自同构群的逆阶，而缺陷 $D_k(\ell)$ 收集余项。对于每个 $k$，加权计数满足一个基于树组分结构的源层的递归，该递归既不涉及组分图列表，也不区分对称与不对称构型，且其指数生成函数是 $\sqrt{1-2x}$ 中的 Laurent 多项式。网络的自同构群是 $2$-群，对于 $k\le5$ 是初等阿贝尔群，但一般情形并非如此。对于 $k\le5$，缺陷是具有一个显著对合的网络之加权计数，它遵循同一递归的扩展。对这两个递归的精确符号求值，在 $k\le5$ 时以闭式形式给出 $|\cP_{\ell,k}|$，其中 $k=5$ 的情形是新的；它重现了 $k\le4$ 的已发表计数，并修正了 $k=3$ 时一个已发表生成函数中的系数。对于每个 $k$，在 $k$ 的显式范围内一致地，我们证明非树-子网络占树-子网络的比例为 $2k(k-1)/\ell$（至首阶），这给出了 $|\cP_{\ell,k}|$ 渐近展开的第三项。我们还证明了网状可见网络超过树-子网络的比例为 $k(k-1)/\ell$，并且 $\cP_{\ell,k}$ 中一个均匀随机网络具有非平凡自同构的概率为 $k(k-1)/(4\ell^3)$（至首阶）。
英文摘要：
Let $\cP_{\ell,k}$ be the set of rooted binary phylogenetic networks with $\ell$ labelled leaves, $k$ reticulations and no parallel edges. We write $|\cP_{\ell,k}|=W_k(\ell)+D_k(\ell)$, where the weighted count $W_k(\ell)$ adds the inverse orders of the leaf-fixing automorphism groups and the defect $D_k(\ell)$ collects the remainder. The weighted count satisfies, for every $k$, a recursion over the source layers of the tree-component structure that involves neither a list of component graphs nor any distinction between symmetric and asymmetric configurations, and its exponential generating function is a Laurent polynomial in $\sqrt{1-2x}$. Automorphism groups of networks are $2$-groups, elementary abelian for $k\le5$ but not in general. For $k\le5$ the defect is the weighted count of networks with a distinguished involution, which obeys an extension of the same recursion. An exact symbolic evaluation of the two recursions yields $|\cP_{\ell,k}|$ in closed form for $k\le5$, the case $k=5$ being new; it reproduces the published counts for $k\le4$ and corrects a coefficient in a published generating function for $k=3$. For every $k$, uniformly over explicit ranges of $k$, we prove that the non-tree-child networks are a fraction $2k(k-1)/\ell$ of the tree-child networks to leading order, which gives the third term of the asymptotic expansion of $|\cP_{\ell,k}|$. We also prove that reticulation-visible networks exceed tree-child networks by the fraction $k(k-1)/\ell$, and that a uniformly random network in $\cP_{\ell,k}$ has a nontrivial automorphism with probability $k(k-1)/(4\ell^3)$ to leading order.
