Bounds for phylogenetic network space metrics

Phylogenetic networks are a generalization of phylogenetic trees that allow for representation of reticulate evolution. Recently, a space of unrooted phylogenetic networks was introduced, where such a network is a connected graph in which every vertex has degree 1 or 3 and whose leaf-set is a fixed set X of taxa. This space, denoted (Formula presented.), is defined in terms of two operations on networks—the nearest neighbor interchange and triangle operations—which can be used to transform any network with leaf set X into any other network with that leaf set. In particular, it gives rise to a metric d on (Formula presented.) which is given by the smallest number of operations required to transform one network in (Formula presented.) into another in (Formula presented.). The metric generalizes the well-known NNI-metric on phylogenetic trees which has been intensively studied in the literature. In this paper, we derive a bound for the metric d as well as a related metric (Formula presented.) which arises when restricting d to the subset of (Formula presented.) consisting of all networks with (Formula presented.) vertices, (Formula presented.). We also introduce two new metrics on networks—the SPR and TBR metrics—which generalize the metrics on phylogenetic trees with the same name and give bounds for these new metrics. We expect our results to eventually have applications to the development and understanding of network search algorithms