  In this paper, we aim to find the conditions for input-state stability (ISS)
and incremental input-state stability ($\delta$ISS) of Gated Graph Neural
Networks (GGNNs). We show that this recurrent version of Graph Neural Networks
(GNNs) can be expressed as a dynamical distributed system and, as a
consequence, can be analysed using model-based techniques to assess its
stability and robustness properties. Then, the stability criteria found can be
exploited as constraints during the training process to enforce the internal
stability of the neural network. Two distributed control examples, flocking and
multi-robot motion control, show that using these conditions increases the
performance and robustness of the gated GNNs.
