  In this paper, we lay the groundwork on the comparison of phylogenetic
networks based on edge contractions and expansions as edit operations, as
originally proposed by Robinson and Foulds to compare trees. We prove that
these operations connect the space of all phylogenetic networks on the same set
of leaves, even if we forbid contractions that create cycles. This allows to
define an operational distance on this space, as the minimum number of
contractions and expansions required to transform one network into another. We
highlight the difference between this distance and the computation of the
maximum common contraction between two networks. Given its ability to outline a
common structure between them, which can provide valuable biological insights,
we study the algorithmic aspects of the latter. We first prove that computing a
maximum common contraction between two networks is NP-hard, even when the
maximum degree, the size of the common contraction, or the number of leaves is
bounded. We also provide lower bounds to the problem based on the
Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time
algorithm for weakly-galled networks, a generalization of galled trees.
