Surprising properties of Maximum Parsimony on phylogenetic networks

Phylogenetic inference aims at reconstructing the evolutionary relationships of different species given some data (e.g. DNA, RNA or proteins). Traditionally, the relationships between species were assumed to be treelike, so the most frequently used phylogenetic inference methods like Maximum Parsimony were originally introduced to reconstruct phylogenetic trees. However, it has been well-known that some evolutionary events like hybridization or horizontal gene transfer cannot be represented by a tree but rather require a phylogenetic network. Therefore, current research seeks to adapt tree inference methods to networks. In the present paper, we analyze Maximum Parsimony on networks for various network definitions which have recently been introduced. For trees, there is a famous result by Tuffley and Steel which states that under a certain model of evolution, Maximum Parsimony always coincides with Maximum Likelihood. We now show that the various definitions Maximum Parsimony on networks can also be proven to be equivalent to certain functions which are similar to the likelihood concept, and we discuss their biological meaningfulness. We also present some complexity results of finding a most parsimonious network.