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We propose a compositional stability analysis methodology for verifying properties of systems that are interconnections of multiple subsystems. The proposed method assembles stability certificates for the interconnected system based on the certificates for the input-output properties of the subsystems. The hierarchy in the analysis is achieved by utilizing dual decomposition ideas in optimization. Decoupled subproblems establish subsystem level input-output properties whereas the ``master'' problem imposes and updates the conditions on the subproblems toward ensuring interconnected system level stability properties. Both global stabilityanalysis and region-of-attraction analysis are discussed.