Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations

Graph neural networks (GNNs) leverage message passing mechanisms to learn the topological features of graph data. Traditional GNNs learns node features in a spatial domain unrelated to the topology, which can hardly ensure topological features. In this paper, we formulates message passing as a system of hyperbolic partial differential equations (hyperbolic PDEs), constituting a dynamical system that explicitly maps node representations into a particular solution space. This solution space is spanned by a set of eigenvectors describing the topological structure of graphs. Within this system, for any moment in time, a node features can be decomposed into a superposition of the basis of eigenvectors. This not only enhances the interpretability of message passing but also enables the explicit extraction of fundamental characteristics about the topological structure. Furthermore, by solving this system of hyperbolic partial differential equations, we establish a connection with spectral graph neural networks (spectral GNNs), serving as a message passing enhancement paradigm for spectral GNNs.We further introduce polynomials to approximate arbitrary filter functions. Extensive experiments demonstrate that the paradigm of hyperbolic PDEs not only exhibits strong flexibility but also significantly enhances the performance of various spectral GNNs across diverse graph tasks.

Graphs reflects relationships between objects of the real world, such as social networks with connections between individuals. Our aim is to learn the feature representations for nodes (i.e., people) through the structure of graphs, which can be useful for downstream applications such as node classification. Modern methods usually learn features of a node by passing messages from its neighboring nodes. This paper uses a mathematical tool named partial differential equation (PDE) to describe the message passing. PDE explains how nodes and their relationships evolve over time. By conducting extensive experiments, we discovered that PDEs have advantages in learning graph structural features, and can significantly enhance the learning capabilities of existing methods.