  Tree-child networks are a recently-described class of directed acyclic graphs
that have risen to prominence in phylogenetics (the study of evolutionary trees
and networks). Although these networks have a number of attractive mathematical
properties, many combinatorial questions concerning them remain intractable. In
this paper, we show that endowing these networks with a biologically relevant
ranking structure yields mathematically tractable objects, which we term ranked
tree-child networks (RTCNs). We explain how to derive exact and explicit
combinatorial results concerning the enumeration and generation of these
networks. We also explore probabilistic questions concerning the properties of
RTCNs when they are sampled uniformly at random. These questions include the
lengths of random walks between the root and leaves (both from the root to the
leaves and from a leaf to the root); the distribution of the 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 that
counts 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.
