# Path Laplacian Encodings for Directed Graphs

arXiv: https://arxiv.org/abs/2610.04657 · HTML: https://mlsanity.com/paper/2610.04657/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 28 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 21 of 21)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [MotifNet: a motif-based Graph Convolutional Network for directed graphs](https://mlsanity.com/paper/1802.01572/) (2018)
- [CktGNN: Circuit Graph Neural Network for Electronic Design Automation](https://mlsanity.com/paper/2308.16406/) (2023)
- [PyG 2.0: Scalable Learning on Real World Graphs](https://mlsanity.com/paper/2507.16991/) (2025)
- [Transformers Meet Directed Graphs](https://mlsanity.com/paper/2302.00049/) (2023)
- [Graph Positional and Structural Encoder](https://mlsanity.com/paper/2307.07107/) (2023)
- [On the Stability of Expressive Positional Encodings for Graphs](https://mlsanity.com/paper/2310.02579/) (2023)
- [High-Level Synthesis Performance Prediction using GNNs: Benchmarking, Modeling, and Advancing](https://mlsanity.com/paper/2201.06848/) (2022)
- [edGNN: a Simple and Powerful GNN for Directed Labeled Graphs](https://mlsanity.com/paper/1904.08745/) (2019)
- [Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes](https://mlsanity.com/paper/2308.06838/) (2023)
- [What Are Good Positional Encodings for Directed Graphs?](https://mlsanity.com/paper/2407.20912/) (2024)
- [What Expressivity Theory Misses: Message Passing Complexity for GNNs](https://mlsanity.com/paper/2509.01254/) (2025)

## Abstract (arXiv)

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.