  Nonlocal boxes are conceptual tools that capture the essence of the
phenomenon of quantum non-locality, central to modern quantum theory and
quantum technologies. We introduce network nonlocal boxes tailored for quantum
networks under the natural assumption that these networks connect independent
sources and do not allow signaling. Hence, these boxes satisfy the No-Signaling
and Independence (NSI) principle. For the case of boxes without inputs,
connecting pairs of bipartite sources and producing binary outputs, we prove
that the sources and boxes producing local random outputs and maximal 2-box
correlations, i.e. $E_2=\sqrt{2}-1$, $E_2^o=1$, are essentially unique.
