  The network structure offers in principle the possibility for novel forms of
quantum nonlocal correlations, that are proper to networks and cannot be traced
back to standard quantum Bell nonlocality. Here we define a notion of genuine
network quantum nonlocality. Our approach is operational and views standard
quantum nonlocality as a resource for producing correlations in networks. We
show several examples of correlations that are genuine network nonlocal,
considering the so-called bilocality network of entanglement swapping. In
particular, we present an example of quantum self-testing which relies on the
network structure; the considered correlations are non-bilocal, but are local
according to the usual definition of Bell locality.
