  A phylogenetic network is a graph-theoretical tool that is used by biologists
to represent the evolutionary history of a collection of species. One potential
way of constructing such networks is via a distance-based approach, where one
is asked to find a phylogenetic network that in some way represents a given
distance matrix, which gives information on the evolutionary distances between
present-day taxa. Here, we consider the following question. For which~$k$ are
unrooted level-$k$ networks uniquely determined by their distance matrices? We
consider this question for shortest distances as well as for the case that the
multisets of all distances is given. We prove that level-$1$ networks and
level-$2$ networks are reconstructible from their shortest distances and
multisets of distances, respectively. Furthermore we show that, in general,
networks of level higher than~$1$ are not reconstructible from shortest
distances and that networks of level higher than~$2$ are not reconstructible
from their multisets of distances.
