  The prevalence of graph-based data has spurred the rapid development of graph
neural networks (GNNs) and related machine learning algorithms. Yet, despite
the many data sets naturally modeled as directed graphs, including citation,
website, and traffic networks, the vast majority of this research focuses on
undirected graphs. In this paper, we propose MagNet, a spectral GNN for
directed graphs based on a complex Hermitian matrix known as the magnetic
Laplacian. This matrix encodes undirected geometric structure in the magnitude
of its entries and directional information in the phase of its entries. A
"charge" parameter attunes spectral information to variation among directed
cycles. We show that MagNet's performance exceeds other spectral GNNs on
directed graph node classification and link prediction tasks for a variety of
datasets and exceeds commonly used spatial GNNs on a majority of such. The
underlying principles of MagNet are such that it can be adapted to other
spectral GNN architectures.
