Title: Learnable Eigenfunctions for Graph Rewiring: Balancing Local Community Structure and Global Connectivity
Primary area: learning on graphs and other geometries & topologies
Keywords: Graph Neural Networks, Graph Rewiring, Spectral Graph Theory, Graph Classification, Node Classification

Abstract: Graph Neural Networks (GNNs) leverage information flow between graph nodes for transductive and inductive tasks. However, default graph topology rarely provides optimal flow for specific tasks, causing over-squashing or over-smoothing. Graph rewiring addresses these issues by altering edges to balance long-range connections (mitigating over-squashing) with locality preservation (preventing over-smoothing). Spectral graph theory offers principled criteria for this trade-off, but has drawbacks: spectral approaches are overly global, and computing spectral quantities lacks scalability. We address these challenges by introducing Inductive Spectral Theory (IST). In IST, spectral quantities and functions are learnable and data-centered, reacting to training data and labels. IST studies spectral elements like the spectral gap and Fiedler vector based on available knowledge. For node and edge-centered tasks, we learn spectral elements from training labels, enabling computation of out-of-sample structural and edgeness measures. This expands structural distances beyond long-range measures like effective resistance to include local intra-cluster-oriented ones. IST is crucial for tasks involving graph populations, such as graph classification, where computing spectral elements is unfeasible, but we learn a consensus spectral space. Our approach strategically adds edges both locally to encourage community structures and globally to facilitate long-range connections while maintaining sparsity. Furthermore, IST serves as a principled graph data augmentation technique, generating diverse training samples that improve model robustness and generalization capabilities. We demonstrate that IST not only improves state-of-the-art graph rewiring performance across multiple benchmarks but also provides a theoretically grounded framework for enhancing GNN architectures through learned spectral properties.

TL;DR: We learn graph eigenfunctions with neural networks to rewire graphs by adding strategic local and global edges, improving GNN performance while serving as effective data augmentation.