  Rearrangement operations transform a phylogenetic tree into another one and
hence induce a metric on the space of phylogenetic trees. Popular operations
for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR
(subtree prune and regraft), and TBR (tree bisection and reconnection).
Recently, these operations have been extended to unrooted phylogenetic
networks, which are generalisations of phylogenetic trees that can model
reticulated evolutionary relationships. Here, we study global and local
properties of spaces of phylogenetic networks under these three operations. In
particular, we prove connectedness and asymptotic bounds on the diameters of
spaces of different classes of phylogenetic networks, including tree-based and
level-k networks. We also examine the behaviour of shortest TBR-sequence
between two phylogenetic networks in a class, and whether the TBR-distance
changes if intermediate networks from other classes are allowed: for example,
the space of phylogenetic trees is an isometric subgraph of the space of
phylogenetic networks under TBR. Lastly, we show that computing the
TBR-distance and the PR-distance of two phylogenetic networks is NP-hard.
