Learning Laplacian Positional Encodings for Heterophilous Graphs
This work addresses a critical limitation in graph neural networks for heterophilous graphs, which are common in real-world networks, representing a significant step in developing more effective positional encodings.
The paper tackles the problem that current graph positional encodings (PEs) can hurt performance on heterophilous graphs, where nearby nodes have different labels, and proposes Learnable Laplacian Positional Encodings (LLPE) to address this, achieving up to 35% and 14% accuracy improvements on synthetic and real-world benchmarks, respectively.
In this work, we theoretically demonstrate that current graph positional encodings (PEs) are not beneficial and could potentially hurt performance in tasks involving heterophilous graphs, where nodes that are close tend to have different labels. This limitation is critical as many real-world networks exhibit heterophily, and even highly homophilous graphs can contain local regions of strong heterophily. To address this limitation, we propose Learnable Laplacian Positional Encodings (LLPE), a new PE that leverages the full spectrum of the graph Laplacian, enabling them to capture graph structure on both homophilous and heterophilous graphs. Theoretically, we prove LLPE's ability to approximate a general class of graph distances and demonstrate its generalization properties. Empirically, our evaluation on 12 benchmarks demonstrates that LLPE improves accuracy across a variety of GNNs, including graph transformers, by up to 35% and 14% on synthetic and real-world graphs, respectively. Going forward, our work represents a significant step towards developing PEs that effectively capture complex structures in heterophilous graphs.