# Cluster Algebras: Network Science and Machine Learning

arXiv: https://arxiv.org/abs/2203.13847 · HTML: https://mlsanity.com/paper/2203.13847/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 80 references that are in the map.
- First posted on arXiv 25 Mar 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Building Quantum Field Theories Out of Neurons](https://mlsanity.com/paper/2112.04527/) (2021)
- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Machine Learning CICY Threefolds](https://mlsanity.com/paper/1806.03121/) (2018)
- [Deep-Learning the Landscape](https://mlsanity.com/paper/1706.02714/) (2017)
- [Learning selection strategies in Buchberger's algorithm](https://mlsanity.com/paper/2005.01917/) (2020)
- [The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning](https://mlsanity.com/paper/1812.02893/) (2018)
- [Quiver Mutations, Seiberg Duality and Machine Learning](https://mlsanity.com/paper/2006.10783/) (2020)
- [Machine Learning Calabi-Yau Four-folds](https://mlsanity.com/paper/2009.02544/) (2020)
- [Learning Algebraic Structures: Preliminary Investigations](https://mlsanity.com/paper/1905.02263/) (2019)
- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Graph Laplacians, Riemannian Manifolds and their Machine-Learning](https://mlsanity.com/paper/2006.16619/) (2020)
- [Neurons on Amoebae](https://mlsanity.com/paper/2106.03695/) (2021)
- [Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves](https://mlsanity.com/paper/2004.05218/) (2020)
- [Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds](https://mlsanity.com/paper/2112.09117/) (2021)
- [Deep Learning Gauss-Manin Connections](https://mlsanity.com/paper/2007.13786/) (2020)
- [Identifying equivalent Calabi--Yau topologies: A discrete challenge from math and physics for machine learning](https://mlsanity.com/paper/2202.07590/) (2022)

## Built on by (most cited first, 9 of 9)

- [Machine Learning Clifford invariants of ADE Coxeter elements](https://mlsanity.com/paper/2310.00041/) (2023)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)
- [Machine Learning Toric Duality in Brane Tilings](https://mlsanity.com/paper/2409.15251/) (2024)
- [Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes](https://mlsanity.com/paper/2411.07467/) (2024)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Machine Learning Mutation-Acyclicity of Quivers](https://mlsanity.com/paper/2411.04209/) (2024)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Learning the Graphical Nature of Symmetries](https://mlsanity.com/paper/2607.12026/) (2026)

## Abstract (arXiv)

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network analysis methods are applied to the exchange graphs for cluster algebras of varying mutation types. The analysis indicates that when the graphs are represented without identifying by permutation equivalence between clusters an elegant symmetry emerges in the quiver exchange graph embedding. The ratio between number of seeds and number of quivers associated to this symmetry is computed for finite Dynkin type algebras up to rank 5, and conjectured for higher ranks. Simple machine learning techniques successfully learn to classify cluster algebras using the data of seeds. The learning performance exceeds 0.9 accuracies between algebras of the same mutation type and between types, as well as relative to artificially generated data.