Learning Laplacian Eigenvectors: a Pre-training Method for Graph Neural Networks (2509.02803v1)

Abstract: We propose a novel framework for pre-training Graph Neural Networks (GNNs) by inductively learning Laplacian eigenvectors. Traditional Message Passing Neural Networks (MPNNs) often struggle to capture global and regional graph structure due to over-smoothing risk as network depth increases. Because the low-frequency eigenvectors of the graph Laplacian matrix encode global information, pre-training GNNs to predict these eigenvectors encourages the network to naturally learn large-scale structural patterns over each graph. Empirically, we show that models pre-trained via our framework outperform baseline models on a variety of graph structure-based tasks. While most existing pre-training methods focus on domain-specific tasks like node or edge feature reconstruction, our self-supervised pre-training framework is structure-based and highly flexible. Eigenvector-learning can be applied to all graph-based datasets, and can be used with synthetic features when task-specific data is sparse.