Tree-child networks are a class of directed acyclic graphs that have recently risen to prominence in phylogenetics. Although these networks have numerous, attractive mathematical properties, many combinatorial questions concerning them remain intractable. We show that endowing tree-child networks with a biologically relevant ranking structure yields mathematically tractable objects, which we term ranked tree-child networks (RTCNs). We derive explicit enumerative formulas and explain how to sample RTCNS uniformly at random. We study the properties of uniform RTCNs, including: lengths of random walks between root and leaves; distribution of number of cherries in the network; and sampling RTCNs conditional on displaying a given tree. We also formulate a conjecture regarding the scaling limit of the process counting the number of lineages in the ancestry of a leaf. The main idea in this paper, namely using ranking as a way to achieve combinatorial tractability, may also extend to other classes of networks.

中文翻译：

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排名树子网络的组合和随机属性

树子网络是一类有向无环图，最近在系统发育学中崭露头角。尽管这些网络具有许多吸引人的数学特性，但许多关于它们的组合问题仍然难以解决。我们表明，赋予树子网络与生物学相关的排名结构会产生数学上易于处理的对象，我们将其称为排名树子网络 (RTCN)。我们推导出显式枚举公式并解释如何随机均匀地采样 RTCNS。我们研究了统一 RTCN 的特性，包括：根和叶之间的随机游走长度；网络中樱桃数量的分布；并以显示给定树为条件对 RTCN 进行采样。我们还提出了一个关于计算叶子祖先谱系数量的过程的缩放限制的猜想。本文的主要思想，即使用排名作为实现组合易处理性的一种方式，也可以扩展到其他类别的网络。