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