We consider a network consisting of n interconnected nonlinear subsystems. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. We use a gain matrix to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, we construct a locally Lipschitz continuous ISS Lyapunov function for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems.