  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 sources and producing binary outputs, we prove that there
is an essentially unique network nonlocal box with local random outputs and
maximal 2-box correlations: $E_2=\sqrt{2}-1$.
