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We consider the problem of stabilizing a class of systems formed by a set of decoupled subsystems (nodes) interconnected through a set of controllers (arcs). Controllers are network-decentralized, i.e., they use information exclusively from the nodes they interconnect. This condition requires a block-structured feedback matrix, having the same structure as the transpose of the overall input matrix of the system. If the subsystems do not have common unstable eigenvalues, we demonstrate that the problem is solvable. In the general case, we provide sufficient conditions for solvability. When subsystems are identical and each input agent controls a pair of subsystems with input matrices having opposite sign (flow networks), we prove that stabilization is possible if and only if the system is connected with the external environment. Our proofs are constructive and lead to structured linear matrix inequalities (LMIs).