  Graph Convolutional Networks (GCNs), similarly to Convolutional Neural
Networks (CNNs), are typically based on two main operations - spatial and
point-wise convolutions. In the context of GCNs, differently from CNNs, a
pre-determined spatial operator based on the graph Laplacian is often chosen,
allowing only the point-wise operations to be learnt. However, learning a
meaningful spatial operator is critical for developing more expressive GCNs for
improved performance. In this paper we propose pathGCN, a novel approach to
learn the spatial operator from random paths on the graph. We analyze the
convergence of our method and its difference from existing GCNs. Furthermore,
we discuss several options of combining our learnt spatial operator with
point-wise convolutions. Our extensive experiments on numerous datasets suggest
that by properly learning both the spatial and point-wise convolutions,
phenomena like over-smoothing can be inherently avoided, and new
state-of-the-art performance is achieved.
