Controlling network systems has become a problem of paramount importance. Optimally controlling a network system with linear dynamics and minimizing a quadratic cost is a particular case of the well-studied linear-quadratic problem. When the specific topology of the network system is ignored, the optimal controller is readily available. However, this results in a emph{centralized} controller, facing limitations in terms of implementation and scalability. Finding the optimal emph{distributed} controller, on the other hand, is intractable in the general case. In this paper, we propose the use of graph neural networks (GNNs) to parametrize and design a distributed controller. GNNs exhibit many desirable properties, such as being naturally distributed and scalable. We cast the distributed linear-quadratic problem as a self-supervised learning problem, which is then used to train the GNN-based controllers. We also obtain sufficient conditions for the resulting closed-loop system to be input-state stable, and derive an upper bound on the trajectory deviation when the system is not accurately known. We run extensive simulations to study the performance of GNN-based distributed controllers and show that they are computationally efficient and scalable.
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