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This paper characterizes a new parametrization of nonlinear networkedincrementally $L_2$-bounded operators in discrete time. The distinctive noveltyis that our parametrization is \emph{free} -- that is, a sparse large-scaleoperator with bounded incremental $L_2$ gain is obtained for any choice of thereal values of our parameters. This property allows one to freely search overoptimal parameters via unconstrained gradient descent, enabling directapplications in large-scale optimal control and system identification. Further,we can embed prior knowledge about the interconnection topology and stabilityproperties of the system directly into the large-scale distributed operator wedesign. Our approach is extremely general in that it can seamlessly encapsulateand interconnect state-of-the-art Neural Network (NN) parametrizations ofstable dynamical systems. To demonstrate the effectiveness of this approach, weprovide a simulation example showcasing the identification of a networkednonlinear system. The results underscore the superiority of our freeparametrizations over standard NN-based identification methods where a priorover the system topology and local stability properties are not enforced.

Unconstrained learning of networked nonlinear systems via free parametrization of stable interconnected operators