  In 2006, Warnow, Evans, Ringe, and Nakhleh proposed a stochastic model
(hereafter, the WERN 2006 model) of multi-state linguistic character evolution
that allowed for homoplasy and borrowing. They proved that if there is no
borrowing between languages and homoplastic states are known in advance, then
the phylogenetic tree of a set of languages is statistically identifiable under
this model, and they presented statistically consistent methods for estimating
these phylogenetic trees. However, they left open the question of whether a
phylogenetic network -- which would explicitly model borrowing between
languages that are in contact -- can be estimated under the model of character
evolution. Here, we establish that under some mild additional constraints on
the WERN 2006 model, the phylogenetic network topology is statistically
identifiable, and we present algorithms to infer the phylogenetic network. We
discuss the ramifications for linguistic phylogenetic network estimation in
practice, and suggest directions for future research.
