  It is known that the Kimura 3ST model of sequence evolution on phylogenetic
trees can be extended quite naturally to arbitrary split systems. However, this
extension relies heavily on mathematical peculiarities of the K3ST model, and
providing an analogous augmentation of the general Markov model has thus far
been elusive. In this paper we rectify this shortcoming by showing how to
extend the general Markov model on trees to to include arbitrary splits; and
even further to more general network models. This is achieved by exploring the
algebra of the generators of the continuous-time Markov chain together with the
"splitting" operator that generates the branching process on phylogenetic
trees. For simplicity we proceed by discussing the two state case and note that
our results are easily extended to more states with little complication.
Intriguingly, upon restriction of the two state general Markov model to the
parameter space of the binary symmetric model, our extension is
indistinguishable from the previous approach only on trees; as soon as any
incompatible splits are introduced the two approaches give rise to differing
probability distributions with disparate structure. Through exploration of a
simple example, we give a tentative argument that our approach to extending to
more general networks has desirable properties that the previous approaches do
not share. In particular, our construction allows for the possibility of
convergent evolution of previously divergent lineages; a property that is of
significant interest for biological applications.
