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 $\mathcal{N}(X)$, 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 $\mathcal N(X)$ which is given by the smallest number of operations required to transform one network in $\mathcal N(X)$ into another in $\mathcal N(X)$. 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 $d_{N\!N\!I}$ which arises when restricting $d$ to the subset of $\mathcal{N}(X)$ consisting of all networks with $2(|X|-1+i)$ vertices, $i \ge 1$. 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.