Path Laplacian Encodings for Directed Graphs

2026-10-03

AI Summary

This study addresses the limitation of existing graph learning methods in fully exploiting edge directionality information in directed graphs. We propose PathLapPE, a spectral positional encoding based on the path Laplacian that captures directed higher-order structural features through a designed direction-aware message passing mechanism. This approach requires no tuning of directionality hyperparameters, integrates seamlessly with standard graph neural network architectures, and maintains computational efficiency. Experimental results demonstrate that PathLapPE significantly enhances model performance on both node classification and graph-level prediction tasks, consistently outperforming existing encoding methods such as the magnetic Laplacian.

Abstract

Directed graphs naturally model many real-world systems in which interactions are asymmetric, such as citation networks, web graphs, and information-flow networks. However, graph learning methods commonly rely on message passing with symmetrized graph representations or positional encodings that only partially exploit edge directionality. We introduce PathLapPE, a novel spectral positional encoding (PE) derived from the path Laplacian on directed graphs. PathLapPE provides node- and edge-level features that encode directional higher-order structure and can be incorporated into standard graph learning architectures. Empirical results on node- and graph-level benchmark tasks show that PathLapPE yields consistent improvements across several architectures, especially when combined with direction-aware message passing. Compared with magnetic Laplacian positional encodings, a widely studied spectral positional encoding for directed graphs, PathLapPE does not require additional fine-tuning of directionality hyperparameters while offering competitive runtime and performance.

Problem

• directed graphs
• positional encoding
• graph learning
• asymmetric interactions
• edge directionality

Innovation

• Directed Graphs
• Path Laplacian
• Spectral Positional Encoding
• Graph Learning
• Higher-order Structure