  Devising and analyzing learning models for spatiotemporal network data is of
importance for tasks including forecasting, anomaly detection, and multi-agent
coordination, among others. Graph Convolutional Neural Networks (GCNNs) are an
established approach to learn from time-invariant network data. The graph
convolution operation offers a principled approach to aggregate multiresolution
information. However, extending the convolution principled learning and
respective analysis to the spatiotemporal domain is challenging because
spatiotemporal data have more intrinsic dependencies. Hence, a higher
flexibility to capture jointly the spatial and the temporal dependencies is
required to learn meaningful higher-order representations. Here, we leverage
product graphs to represent the spatiotemporal dependencies in the data and
introduce Graph-Time Convolutional Neural Networks (GTCNNs) as a principled
architecture to aid learning. The proposed approach can work with any type of
product graph and we also introduce a parametric product graph to learn also
the spatiotemporal coupling. The convolution principle further allows a similar
mathematical tractability as for GCNNs. In particular, the stability result
shows GTCNNs are stable to spatial perturbations but there is an implicit
trade-off between discriminability and robustness; i.e., the more complex the
model, the less stable. Extensive numerical results on benchmark datasets
corroborate our findings and show the GTCNN compares favorably with
state-of-the-art solutions. We anticipate the GTCNN to be a starting point for
more sophisticated models that achieve good performance but are also
fundamentally grounded.
