  Most existing graph neural networks (GNNs) are limited to undirected graphs,
whose restricted scope of the captured relational information hinders their
expressive capabilities and deployments in real-world scenarios. Compared with
undirected graphs, directed graphs (digraphs) fit the demand for modeling more
complex topological systems by capturing more intricate relationships between
nodes, such as formulating transportation and financial networks. While some
directed GNNs have been introduced, their inspiration mainly comes from deep
learning architectures, which lead to redundant complexity and computation,
making them inapplicable to large-scale databases. To address these issues, we
propose LightDiC, a scalable variant of the digraph convolution based on the
magnetic Laplacian. Since topology-related computations are conducted solely
during offline pre-processing, LightDiC achieves exceptional scalability,
enabling downstream predictions to be trained separately without incurring
recursive computational costs. Theoretical analysis shows that LightDiC
utilizes directed information to achieve message passing based on the complex
field, which corresponds to the proximal gradient descent process of the
Dirichlet energy optimization function from the perspective of digraph signal
denoising, ensuring its expressiveness. Experimental results demonstrate that
LightDiC performs comparably well or even outperforms other SOTA methods in
various downstream tasks, with fewer learnable parameters and higher training
efficiency. Notably, LightDiC is the first DiGNN to provide satisfactory
results in the most representative large-scale database (ogbn-papers100M).
