Towards Stable, Globally Expressive Graph Representations with Laplacian Eigenvectors

Graph neural networks (GNNs) have achieved remarkable success in a variety of machine learning tasks over graph data. Existing GNNs usually rely on message passing, i.e., computing node representations by gathering information from the neighborhood, to build their underlying computational graphs. Such an approach has been shown fairly limited in expressive power, and often fails to capture global characteristics of graphs. To overcome the issue, a popular solution is to use Laplacian eigenvectors as additional node features, as they are known to contain global positional information of nodes, and can serve as extra node identifiers aiding GNNs to separate structurally similar nodes. Since eigenvectors naturally come with symmetries---namely, $O(p)$-group symmetry for every $p$ eigenvectors with equal eigenvalue, properly handling such symmetries is crucial for the stability and generalizability of Laplacian eigenvector augmented GNNs. However, using a naive $O(p)$-group invariant encoder for each $p$-dimensional eigenspace may not keep the full expressivity in the Laplacian eigenvectors. Moreover, computing such invariants inevitably entails a hard split of Laplacian eigenvalues according to their numerical identity, which suffers from great instability when the graph structure has small perturbations. In this paper, we propose a novel method exploiting Laplacian eigenvectors to generate stable and globally expressive graph representations. The main difference from previous works is that (i) our method utilizes learnable $O(p)$-invariant representations for each Laplacian eigenspace of dimension $p$, which are built upon powerful orthogonal group equivariant neural network layers already well studied in the literature, and that (ii) our method deals with numerically close eigenvalues in a smooth fashion, ensuring its better robustness against perturbations. Experiments on various graph learning benchmarks witness the competitive performance of our method, especially its great potential to learn global properties of graphs.