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.