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We analyse the stability of large-scale nonlinear stochastic systems using appropriate stochastic passivity properties of their subsystems and the structure of their interactions. Stochastic stability, and noise-to-state stability, of the network is established from the diagonal stability of a dissipativity matrix that incorporates information about the passivity properties of the subsystems and their interconnection. Next, equilibrium-independent conditions for the verification of the relevant passivity properties of the subsystems are derived. Finally, the decomposition-based approach is illustrated on a class of biological reaction networks.