Title: Achieving Stable and Globally Expressive Graph Representations via Laplacian Eigenvectors

Abstract:

Integrating Laplacian eigenvectors as supplementary node features has emerged as a prevalent strategy for enhancing the expressive capacity of Graph Neural Networks (GNNs), owing to their dual role as structural identifiers and global node coordinates. However, ensuring the stability and generalizability of these augmented GNNs hinges critically on the proper management of orthogonal group symmetry among the eigenvectors. Prior research indicates that employing a straightforward $O(p)$-group invariant encoder for every $p$-dimensional eigenspace frequently results in both numerical instability and a reduction in expressivity.

To address these challenges, this study introduces an innovative approach that leverages Laplacian eigenvectors to construct graph representations that are both globally expressive and stable. Our methodology diverges from existing techniques in two key aspects: (i) it employs learnable $O(p)$-invariant representations for each Laplacian eigenspace of dimension $p$. These representations are grounded in robust orthogonal group equivariant neural network layers, which have been extensively investigated in prior literature; and (ii) it handles numerically proximate eigenvalues in a smooth manner, thereby significantly enhancing robustness to perturbations. Empirical evaluations across a range of graph learning benchmarks demonstrate that our method achieves competitive performance, highlighting its substantial potential for capturing the global characteristics of graphs.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC