  We present a generalization of graph convolutional networks by generalizing
the diffusion operation underlying this class of graph neural networks. These
sheaf neural networks are based on the sheaf Laplacian, a generalization of the
graph Laplacian that encodes additional relational structure parameterized by
the underlying graph. The sheaf Laplacian and associated matrices provide an
extended version of the diffusion operation in graph convolutional networks,
providing a proper generalization for domains where relations between nodes are
non-constant, asymmetric, and varying in dimension. We show that the resulting
sheaf neural networks can outperform graph convolutional networks in domains
where relations between nodes are asymmetric and signed.
