Nonlocal boxes for networks

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