  The study of entanglement in multipartite quantum states plays a major role
in quantum information theory and genuine multipartite entanglement signals one
of its strongest forms for applications. However, its characterization for
general (mixed) states is a highly nontrivial problem. We introduce a
particularly simple subclass of multipartite states, which we term
pair-entangled network (PEN) states, as those that can be created by
distributing exclusively bipartite entanglement in a connected network. We show
that genuine multipartite entanglement in a PEN state depends on both the level
of noise and the network topology and, in sharp contrast to the case of pure
states, it is not guaranteed by the mere distribution of mixed bipartite
entangled states. Our main result is a markedly drastic feature of this
phenomenon: the amount of connectivity in the network determines whether
genuine multipartite entanglement is robust to noise for any system size or
whether it is completely washed out under the slightest form of noise for a
sufficiently large number of parties. This latter case implies fundamental
limitations for the application of certain networks in realistic scenarios,
where the presence of some form of noise is unavoidable. To illustrate the
applicability of PEN states to study the complex phenomenology behind
multipartite entanglement, we also use them to prove superactivation of genuine
multipartite nonlocality for any number of parties.
