GrassNet: State Space Model Meets Graph Neural Network

TL;DR

GrassNet tackles limitations of traditional spectral GNN filters that rely on predefined polynomial or wavelet bases, which struggle when the graph spectrum is concentrated or contains repeated eigenvalues. It introduces graph state space networks to model the entire spectrum sequence and produce per-frequency filters, enabling differentiated treatment of numerically identical frequencies. The approach provides theoretical and empirical evidence of greater expressive power, along with an efficient and robust training procedure, validated on nine public benchmarks where GrassNet often achieves top results. This work establishes a new spectral filtering paradigm for GNNs with potential impact on real-world graph learning tasks requiring nuanced spectral modulation.

Abstract

Designing spectral convolutional networks is a formidable task in graph learning. In traditional spectral graph neural networks (GNNs), polynomial-based methods are commonly used to design filters via the Laplacian matrix. In practical applications, however, these polynomial methods encounter inherent limitations, which primarily arise from the the low-order truncation of polynomial filters and the lack of overall modeling of the graph spectrum. This leads to poor performance of existing spectral approaches on real-world graph data, especially when the spectrum is highly concentrated or contains many numerically identical values, as they tend to apply the exact same modulation to signals with the same frequencies. To overcome these issues, in this paper, we propose Graph State Space Network (GrassNet), a novel graph neural network with theoretical support that provides a simple yet effective scheme for designing and learning arbitrary graph spectral filters. In particular, our GrassNet introduces structured state space models (SSMs) to model the correlations of graph signals at different frequencies and derives a unique rectification for each frequency in the graph spectrum. To the best of our knowledge, our work is the first to employ SSMs for the design of GNN spectral filters, and it theoretically offers greater expressive power compared with polynomial filters. Extensive experiments on nine public benchmarks reveal that GrassNet achieves superior performance in real-world graph modeling tasks.

Figures (6)

• Figure 1: Spectral decomposition of a toy graph. The values of the eigenvector corresponding to each eigenvalue are represented by the shading of the nodes, and the number on each node indicates its category.
• Figure 2: Visualization of some connected components within three graph benchmarks.
• Figure 3: (a-c): The kernel density estimation (KDE) on Cora, CiteSeer and Photo. (d-e): The classification accuracy of GCN and BernNet on the original graph topology and two modified graph topologies.
• Figure 4: Overall framework of the proposed GrassNet. The SSM graph filter is composed of cascaded SSM layers, which takes the ordered graph spectrum as input and outputs a filtering coefficient for each frequency within the spectrum. With the learned filtering coefficients, graph convolution is then applied to the latent embeddings of nodes, derived from fully connected layers.
• Figure 5: Illustration of filters learned from real-world datasets by GrassNet.