# Generalized Laplacian Positional Encoding for Graph Representation Learning

arXiv: https://arxiv.org/abs/2210.15956 · HTML: https://mlsanity.com/paper/2210.15956/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 28 Oct 2022.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [A Generalization of Transformer Networks to Graphs](https://mlsanity.com/paper/2012.09699/) (2020)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [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)
- [Labeling Trick: A Theory of Using Graph Neural Networks for Multi-Node Representation Learning](https://mlsanity.com/paper/2010.16103/) (2020)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks](https://mlsanity.com/paper/2203.00199/) (2022)
- [p-Laplacian Based Graph Neural Networks](https://mlsanity.com/paper/2111.07337/) (2021)

## Built on by (most cited first, 21 of 21)

- [Swallowing the Bitter Pill: Simplified Scalable Conformer Generation](https://mlsanity.com/paper/2311.17932/) (2023)
- [On the Stability of Expressive Positional Encodings for Graphs](https://mlsanity.com/paper/2310.02579/) (2023)
- [Graph Positional Encoding via Random Feature Propagation](https://mlsanity.com/paper/2303.02918/) (2023)
- [NExT-Mol: 3D Diffusion Meets 1D Language Modeling for 3D Molecule Generation](https://mlsanity.com/paper/2502.12638/) (2025)
- [On the Expressive Power of Spectral Invariant Graph Neural Networks](https://mlsanity.com/paper/2406.04336/) (2024)
- [What Can We Learn from State Space Models for Machine Learning on Graphs?](https://mlsanity.com/paper/2406.05815/) (2024)
- [Manifold Diffusion Fields](https://mlsanity.com/paper/2305.15586/) (2023)
- [Best of Both Worlds: Advantages of Hybrid Graph Sequence Models](https://mlsanity.com/paper/2411.15671/) (2024)
- [Graph Representational Learning: When Does More Expressivity Hurt Generalization?](https://mlsanity.com/paper/2505.11298/) (2025)
- [SMAGDi: Socratic Multi Agent Interaction Graph Distillation for Efficient High Accuracy Reasoning](https://mlsanity.com/paper/2511.05528/) (2025)
- [Understanding and Improving Laplacian Positional Encodings For Temporal GNNs](https://mlsanity.com/paper/2506.01596/) (2025)
- [Simulation as Supervision: Mechanistic Pretraining for Scientific Discovery](https://mlsanity.com/paper/2507.08977/) (2025)
- [Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networks](https://mlsanity.com/paper/2509.24886/) (2025)
- [Weisfeiler and Leman Go Loopy: A New Hierarchy for Graph Representational Learning](https://mlsanity.com/paper/2403.13749/) (2024)
- [LEAP: Local ECT-Based Learnable Positional Encodings for Graphs](https://mlsanity.com/paper/2510.00757/) (2025)
- [Positional Encoding meets Persistent Homology on Graphs](https://mlsanity.com/paper/2506.05814/) (2025)
- [Nonlinear Laplacians Improve Signed-Directed Graph Learning](https://mlsanity.com/paper/2608.00836/) (2026)
- [Learning from Historical Activations in Graph Neural Networks](https://mlsanity.com/paper/2601.01123/) (2026)
- [Bridging Distance and Spectral Positional Encodings via Anchor-Based Diffusion Geometry Approximation](https://mlsanity.com/paper/2601.04517/) (2026)
- [Bridging Local and Global Knowledge: Cascaded Mixture-of-Experts Learning for Near-Shortest Path Routing](https://mlsanity.com/paper/2603.15541/) (2026)
- [Beyond Modality Fusion: Deep Ensembles for Multimodal Classification](https://mlsanity.com/paper/2607.05019/) (2026)

## Abstract (arXiv)

Graph neural networks (GNNs) are the primary tool for processing graph-structured data. Unfortunately, the most commonly used GNNs, called Message Passing Neural Networks (MPNNs) suffer from several fundamental limitations. To overcome these limitations, recent works have adapted the idea of positional encodings to graph data. This paper draws inspiration from the recent success of Laplacian-based positional encoding and defines a novel family of positional encoding schemes for graphs. We accomplish this by generalizing the optimization problem that defines the Laplace embedding to more general dissimilarity functions rather than the 2-norm used in the original formulation. This family of positional encodings is then instantiated by considering p-norms. We discuss a method for calculating these positional encoding schemes, implement it in PyTorch and demonstrate how the resulting positional encoding captures different properties of the graph. Furthermore, we demonstrate that this novel family of positional encodings can improve the expressive power of MPNNs. Lastly, we present preliminary experimental results.