Identifiability of tree-child phylogenetic networks under a probabilistic recombination-mutation model of evolution

Phylogenetic networks are an extension of phylogenetic trees which are
used to represent evolutionary histories in which reticulation events (such as recombination and hybridization) have occurred. A central question for such networks is that of identifiability, which essentially asks under what circumstances can we reliably identify the phylogenetic network that gave rise to the observed data? Recently, identifiability results have appeared for networks relative to a model of sequence evolution that generalizes the standard Markov models used for phylogenetic trees. However, these results are quite limited in terms of the complexity of the networks that are considered. In this paper, by introducing an alternative probabilistic model for evolution along a network that is based on some ground-breaking work by Thatte for pedigrees, we are able to obtain an identifiability result for a much larger class of phylogenetic networks (essentially the class of so-called tree-child networks). To prove our main theorem, we derive some new results for identifying tree-child networks combinatorially, and then adapt some techniques developed by Thatte for pedigrees to show that our combinatorial results imply identifiability in the probabilistic setting. We hope that the introduction of our new model for networks could lead to new approaches to reliably construct phylogenetic networks.