水平$k$系统发生网络的分支过程方法
A branching process approach to level-$k$ phylogenetic networks
摘要：
随机系统发育网络的数学分析，通过分析和算法的方法，在过去的几年中受到越来越多的关注。在本工作中，我们引入分支过程的方法来研究它们。在这种情况下，这种做法似乎是新的。我们的主要结果集中在随机水平$k$网络与$n$标记的叶子。虽然在这样的网络中的网状顶点的数量通常是线性的$n$，我们证明了他们的渐近的全球和当地的形状是树状在一个定义明确的意义。我们证明了一个大网络中的顶点的深度过程在经过$n^{-1/2}$重新标度后收敛于布朗漂移。我们还建立了大型随机水平$k$网络向一个新的随机无限网络的Benjamini-Schramm收敛性。
英文摘要：
The mathematical analysis of random phylogenetic networks via analytic and algorithmic methods has received increasing attention in the past years. In the present work we introduce branching process methods to their study. This approach appears to be new in this context. Our main results focus on random level-$k$ networks with $n$ labelled leaves. Although the number of reticulation vertices in such networks is typically linear in $n$, we prove that their asymptotic global and local shape is tree-like in a well-defined sense. We show that the depth process of vertices in a large network converges towards a Brownian excursion after rescaling by $n^{-1/2}$. We also establish Benjamini--Schramm convergence of large random level-$k$ networks towards a novel random infinite network.
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