# Effective Structural Encodings via Local Curvature Profiles

arXiv: https://arxiv.org/abs/2311.14864 · HTML: https://mlsanity.com/paper/2311.14864/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 53 references that are in the map.
- First posted on arXiv 24 Nov 2023.

## Builds on (most cited first, 30 of 31)

- [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)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Graph Neural Networks in Recommender Systems: A Survey](https://mlsanity.com/paper/2011.02260/) (2020)
- [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)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Modeling polypharmacy side effects with graph convolutional networks](https://mlsanity.com/paper/1802.00543/) (2018)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness](https://mlsanity.com/paper/2110.03753/) (2021)
- [GraphiT: Encoding Graph Structure in Transformers](https://mlsanity.com/paper/2106.05667/) (2021)
- [FoSR: First-order spectral rewiring for addressing oversquashing in GNNs](https://mlsanity.com/paper/2210.11790/) (2022)
- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [GRPE: Relative Positional Encoding for Graph Transformer](https://mlsanity.com/paper/2201.12787/) (2022)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Mitigating Over-Smoothing and Over-Squashing using Augmentations of Forman-Ricci Curvature](https://mlsanity.com/paper/2309.09384/) (2023)
- [Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows](https://mlsanity.com/paper/1607.08654/) (2016)
- [Curvature-based Clustering on Graphs](https://mlsanity.com/paper/2307.10155/) (2023)

## Built on by (most cited first, 13 of 13)

- [Balancing Efficiency and Expressiveness: Subgraph GNNs with Walk-Based Centrality](https://mlsanity.com/paper/2501.03113/) (2025)
- [Performance Heterogeneity in Graph Neural Networks: Lessons for Architecture Design and Preprocessing](https://mlsanity.com/paper/2503.00547/) (2025)
- [Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures](https://mlsanity.com/paper/2510.04936/) (2025)
- [Enhancing the Utility of Higher-Order Information in Relational Learning](https://mlsanity.com/paper/2502.09570/) (2025)
- [A Geometric Perspective on the Difficulties of Learning GNN-based SAT Solvers](https://mlsanity.com/paper/2508.21513/) (2025)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)
- [Neural Algorithmic Reasoning for Approximate k-Coloring with Recursive Warm Starts](https://mlsanity.com/paper/2601.05137/) (2026)
- [ASEHybrid: When Geometry Matters Beyond Homophily in Graph Neural Networks](https://mlsanity.com/paper/2601.18912/) (2026)
- [Sobolev--Ricci Curvature](https://mlsanity.com/paper/2603.12652/) (2026)
- [Periodic Topological Deep Learning for Polymer Design and Discovery](https://mlsanity.com/paper/2605.26833/) (2026)
- [VeriDrive: Verifiable Counterfactual Supervision for Cost-Efficient Vision-Language Planning](https://mlsanity.com/paper/2606.07338/) (2026)
- [Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature](https://mlsanity.com/paper/2607.22381/) (2026)
- [Look Before You Lift: Visual and Quantitative Diagnostics for Topological Deep Learning](https://mlsanity.com/paper/2608.15388/) (2026)

## Abstract (arXiv)

Structural and Positional Encodings can significantly improve the performance of Graph Neural Networks in downstream tasks. Recent literature has begun to systematically investigate differences in the structural properties that these approaches encode, as well as performance trade-offs between them. However, the question of which structural properties yield the most effective encoding remains open. In this paper, we investigate this question from a geometric perspective. We propose a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and show that it significantly outperforms existing encoding approaches. We further show that combining local structural encodings, such as LCP, with global positional encodings improves downstream performance, suggesting that they capture complementary geometric information. Finally, we compare different encoding types with (curvature-based) rewiring techniques. Rewiring has recently received a surge of interest due to its ability to improve the performance of Graph Neural Networks by mitigating over-smoothing and over-squashing effects. Our results suggest that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring.