  Transmission through a complex network of nonlinear one-dimensional leads is
discussed by extending the stationary scattering theory on quantum graphs to
the nonlinear regime. We show that the existence of cycles inside the graph
leads to a large number of sharp resonances that dominate scattering. The
latter resonances are then shown to be extremely sensitive to the nonlinearity
and display multi-stability and hysteresis. This work provides a framework for
the study of light propagation in complex optical networks.
