The generation time is commonly defined as the mean age of mothers at birth. In matrix population models, a general formula is available to compute this quantity. However, it is complex and hard to interpret. Here, we present a new approach where the generation time is envisioned as a return time in an appropriate Markov chain. This yields surprisingly simple results, such as the fact that the generation time is the inverse of the sum of the elasticities of the growth rate to changes in the fertilities. This result sheds new light on the interpretation of elasticities (which as we show correspond to the frequency of events in the ancestral lineage of the population), and we use it to generalize a result known as Lebreton's formula. Finally, we also show that the generation time can be seen as a random variable, and we give a general expression for its distribution.

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.

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.