Graph Wave Networks

2025-04-22

The Web Conference

AI Summary

Existing graph neural networks (GNNs) model message passing as heat diffusion, governed by the heat equation involving a first-order time derivative. This formulation struggles to capture signal oscillations on graphs and suffers from poor numerical stability, leading to over-smoothing and heterophily issues. To address these limitations, this work pioneers a wave-propagation perspective for graph message passing, introducing the Graph Wave Equation—formulated via the graph Laplacian and incorporating a second-order time derivative. We establish its theoretical connection to spectral graph neural networks. Furthermore, we design explicit and implicit numerical solvers, a spectral-domain wave propagation mechanism, and multi-order graph convolutions. The proposed framework significantly improves numerical stability and training efficiency, achieves state-of-the-art performance across multiple benchmark datasets, and demonstrates strong robustness against over-smoothing and on heterophilous graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Dynamics modeling has been introduced as a novel paradigm in message passing (MP) of graph neural networks (GNNs). Existing methods consider MP between nodes as a heat diffusion process, and leverage heat equation to model the temporal evolution of nodes in the embedding space. However, heat equation can hardly depict the wave nature of graph signals in graph signal processing. Besides, heat equation is essentially a partial differential equation (PDE) involving a first partial derivative of time, whose numerical solution usually has low stability, and leads to inefficient model training. In this paper, we would like to depict more wave details in MP, since graph signals are essentially wave signals that can be seen as a superposition of a series of waves in the form of eigenvector. This motivates us to consider MP as a wave propagation process to capture the temporal evolution of wave signals in the space. Based on wave equation in physics, we innovatively develop a graph wave equation to leverage the wave propagation on graphs. In details, we demonstrate that the graph wave equation can be connected to traditional spectral GNNs, facilitating the design of graph wave networks (GWNs) based on various Laplacians and enhancing the performance of the spectral GNNs. Besides, the graph wave equation is particularly a PDE involving a second partial derivative of time, which has stronger stability on graphs than the heat equation that involves a first partial derivative of time. Additionally, we theoretically prove that the numerical solution derived from the graph wave equation are constantly stable, enabling to significantly enhance model efficiency while ensuring its performance. Extensive experiments show that GWNs achieve state-of-the-art and efficient performance on benchmark datasets, and exhibit outstanding performance in addressing challenging graph problems, such as over-smoothing and heterophily. Our code is available at https://github.com/YueAWu/Graph-Wave-Networks.

Problem

• Model wave nature of graph signals in message passing
• Enhance stability and efficiency of graph neural networks
• Address over-smoothing and heterophily in graph learning

Innovation

• Graph wave equation replaces heat diffusion
• Second-order PDE enhances numerical stability
• Connects to spectral GNNs via Laplacians