  There recently has been a surge of interest in developing a new class of deep
learning (DL) architectures that integrate an explicit time dimension as a
fundamental building block of learning and representation mechanisms. In turn,
many recent results show that topological descriptors of the observed data,
encoding information on the shape of the dataset in a topological space at
different scales, that is, persistent homology of the data, may contain
important complementary information, improving both performance and robustness
of DL. As convergence of these two emerging ideas, we propose to enhance DL
architectures with the most salient time-conditioned topological information of
the data and introduce the concept of zigzag persistence into time-aware graph
convolutional networks (GCNs). Zigzag persistence provides a systematic and
mathematically rigorous framework to track the most important topological
features of the observed data that tend to manifest themselves over time. To
integrate the extracted time-conditioned topological descriptors into DL, we
develop a new topological summary, zigzag persistence image, and derive its
theoretical stability guarantees. We validate the new GCNs with a time-aware
zigzag topological layer (Z-GCNETs), in application to traffic forecasting and
Ethereum blockchain price prediction. Our results indicate that Z-GCNET
outperforms 13 state-of-the-art methods on 4 time series datasets.
