Nonlocal boxes are conceptual tools that capture the essence of the phenomenon of quantum nonlocality, 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 principle. For the case of boxes without inputs, connecting pairs of bipartite sources and producing binary outputs, we prove that the sources root and boxes producing local random outputs and maximal two-box correlations, i.e., E2 = 2 - 1, E2o = 1, are essentially unique.