# Graph Positional and Structural Encoder

arXiv: https://arxiv.org/abs/2307.07107 · HTML: https://mlsanity.com/paper/2307.07107/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 98 references that are in the map.
- First posted on arXiv 14 Jul 2023.

## Builds on (most cited first, 30 of 61)

- [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)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Graph Contrastive Learning with Augmentations](https://mlsanity.com/paper/2010.13902/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deep Graph Infomax](https://mlsanity.com/paper/1809.10341/) (2018)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [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)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [GraphMAE: Self-Supervised Masked Graph Autoencoders](https://mlsanity.com/paper/2205.10803/) (2022)
- [Residual Gated Graph ConvNets](https://mlsanity.com/paper/1711.07553/) (2017)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [OGB-LSC: A Large-Scale Challenge for Machine Learning on Graphs](https://mlsanity.com/paper/2103.09430/) (2021)

## Built on by (most cited first, 30 of 33)

- [Relational Graph Transformer](https://mlsanity.com/paper/2505.10960/) (2025)
- [Learning Graph Quantized Tokenizers](https://mlsanity.com/paper/2410.13798/) (2024)
- [Bringing Graphs to the Table: Zero-shot Node Classification via Tabular Foundation Models](https://mlsanity.com/paper/2509.07143/) (2025)
- [Deep Reinforcement Learning for Autonomous Cyber Defence: A Survey](https://mlsanity.com/paper/2310.07745/) (2023)
- [Towards A Universal Graph Structural Encoder](https://mlsanity.com/paper/2504.10917/) (2025)
- [Towards Foundation Models on Graphs: An Analysis on Cross-Dataset Transfer of Pretrained GNNs](https://mlsanity.com/paper/2412.17609/) (2024)
- [HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations](https://mlsanity.com/paper/2505.15405/) (2025)
- [View Space: Learning Representation across Arbitrary Graphs](https://mlsanity.com/paper/2512.11561/) (2025)
- [Towards Graph Foundation Models: A Study on the Generalization of Positional and Structural Encodings](https://mlsanity.com/paper/2412.07407/) (2024)
- [Towards Invariance to Node Identifiers in Graph Neural Networks](https://mlsanity.com/paper/2502.13660/) (2025)
- [Learning Laplacian Positional Encodings for Heterophilous Graphs](https://mlsanity.com/paper/2504.20430/) (2025)
- [Graph Positional Autoencoders as Self-supervised Learners](https://mlsanity.com/paper/2505.23345/) (2025)
- [Bridging Input Feature Spaces Towards Graph Foundation Models](https://mlsanity.com/paper/2605.04834/) (2026)
- [Universality and Approximation Rates of Graph Neural Networks with Random Features](https://mlsanity.com/paper/2607.26699/) (2026)
- [On the Utilization of Unique Node Identifiers in Graph Neural Networks](https://mlsanity.com/paper/2411.02271/) (2024)
- [Graph Alignment for Benchmarking Graph Neural Networks and Learning Positional Encodings](https://mlsanity.com/paper/2505.13087/) (2025)
- [Optimizing Ego Vehicle Trajectory Prediction: The Graph Enhancement Approach](https://mlsanity.com/paper/2312.13104/) (2023)
- [ExGRG: Explicitly-Generated Relation Graph for Self-Supervised Representation Learning](https://mlsanity.com/paper/2402.06737/) (2024)
- [TopER: Topological Embeddings in Graph Representation Learning](https://mlsanity.com/paper/2410.01778/) (2024)
- [Simple Path Structural Encoding for Graph Transformers](https://mlsanity.com/paper/2502.09365/) (2025)
- [GReAT: leveraging geometric artery data to improve wall shear stress assessment](https://mlsanity.com/paper/2508.19030/) (2025)
- [The Role of Node Features in Graph Pooling](https://mlsanity.com/paper/2605.06250/) (2026)
- [Range-aware Positional Encoding via High-order Pretraining: Theory and Practice](https://mlsanity.com/paper/2409.19117/) (2024)
- [LASE: Learned Adjacency Spectral Embeddings](https://mlsanity.com/paper/2412.17734/) (2024)
- [Ken Utilization Layer: Hebbian Replay Within a Student's Ken for Adaptive Exercise Recommendation](https://mlsanity.com/paper/2507.00032/) (2025)
- [A Graph Laplacian Eigenvector-based Pre-training Method for Graph Neural Networks](https://mlsanity.com/paper/2509.02803/) (2025)
- [On The Expressive Power of GNN Derivatives](https://mlsanity.com/paper/2510.02565/) (2025)
- [Graph Learning Should Move Beyond Restrictive Views of Spectral and Message-Passing GNNs](https://mlsanity.com/paper/2602.10031/) (2026)
- [Communicability-Inspired Positional Encoding (CIPE)](https://mlsanity.com/paper/2606.25293/) (2026)
- [On Preserving Geometrical Invariance for Superpixel Image Classification using Graph Transformer](https://mlsanity.com/paper/2607.04262/) (2026)

## Abstract (arXiv)

Positional and structural encodings (PSE) enable better identifiability of nodes within a graph, rendering them essential tools for empowering modern GNNs, and in particular graph Transformers. However, designing PSEs that work optimally for all graph prediction tasks is a challenging and unsolved problem. Here, we present the Graph Positional and Structural Encoder (GPSE), the first-ever graph encoder designed to capture rich PSE representations for augmenting any GNN. GPSE learns an efficient common latent representation for multiple PSEs, and is highly transferable: The encoder trained on a particular graph dataset can be used effectively on datasets drawn from markedly different distributions and modalities. We show that across a wide range of benchmarks, GPSE-enhanced models can significantly outperform those that employ explicitly computed PSEs, and at least match their performance in others. Our results pave the way for the development of foundational pre-trained graph encoders for extracting positional and structural information, and highlight their potential as a more powerful and efficient alternative to explicitly computed PSEs and existing self-supervised pre-training approaches. Our framework and pre-trained models are publicly available at https://github.com/G-Taxonomy-Workgroup/GPSE. For convenience, GPSE has also been integrated into the PyG library to facilitate downstream applications.