  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.
