Cluster Algebras: Network Science and Machine Learning

TL;DR

This paper explores the application of network science and machine learning to cluster algebras, revealing symmetries in exchange graphs and achieving high classification accuracy of algebra types.

Contribution

It introduces a novel analysis of cluster algebras using network and machine learning techniques, uncovering symmetries and demonstrating effective classification methods.

Findings

Symmetry in quiver exchange graph embeddings without permutation identification.

High classification accuracy (>0.9) for cluster algebra types using machine learning.

Computed and conjectured ratios of seeds to quivers for finite Dynkin types.

Abstract

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.