Range-aware Positional Encoding via High-order Pretraining: Theory and Practice

Unsupervised pre-training on vast amounts of graph data is critical in real-world applications wherein labeled data is limited, such as molecule properties prediction or materials science. Existing approaches pre-train models for specific graph domains, neglecting the inherent connections within networks. This limits their ability to transfer knowledge to various supervised tasks. In this work, we propose a novel pre-training strategy on graphs that focuses on modeling their multi-resolution structural information, allowing us to capture global information of the whole graph while preserving local structures around its nodes. We extend the work of Wave}let Positional Encoding (WavePE) from (Ngo et al., 2023) by pretraining a High-Order Permutation-Equivariant Autoencoder (HOPE-WavePE) to reconstruct node connectivities from their multi-resolution wavelet signals. Unlike existing positional encodings, our method is designed to become sensitivity to the input graph size in downstream tasks, which efficiently capture global structure on graphs. Since our approach relies solely on the graph structure, it is also domain-agnostic and adaptable to datasets from various domains, therefore paving the wave for developing general graph structure encoders and graph foundation models. We theoretically demonstrate that there exists a parametrization of such architecture that it can predict the output adjacency up to arbitrarily low error. We also evaluate HOPE-WavePE on graph-level prediction tasks of different areas and show its superiority compared to other methods.