  Graph processes exhibit a temporal structure determined by the sequence index
and and a spatial structure determined by the graph support. To learn from
graph processes, an information processing architecture must then be able to
exploit both underlying structures. We introduce Graph Recurrent Neural
Networks (GRNNs) as a general learning framework that achieves this goal by
leveraging the notion of a recurrent hidden state together with graph signal
processing (GSP). In the GRNN, the number of learnable parameters is
independent of the length of the sequence and of the size of the graph,
guaranteeing scalability. We prove that GRNNs are permutation equivariant and
that they are stable to perturbations of the underlying graph support. To
address the problem of vanishing gradients, we also put forward gated GRNNs
with three different gating mechanisms: time, node and edge gates. In numerical
experiments involving both synthetic and real datasets, time-gated GRNNs are
shown to improve upon GRNNs in problems with long term dependencies, while node
and edge gates help encode long range dependencies present in the graph. The
numerical results also show that GRNNs outperform GNNs and RNNs, highlighting
the importance of taking both the temporal and graph structures of a graph
process into account.
