Cluster Algebras: Network Science and Machine Learning

TL;DR

This work applies network science and supervised learning to cluster algebras, revealing a robust symmetry in quiver embeddings when permutation equivalence is not imposed and uncovering integer seed-to-quiver ratios for finite Dynkin types. By analyzing seed and quiver exchange graphs up to depth 4 across finite, finite-mutation, and infinite types, the authors show distinct network-analytic signatures and identify a strong, intrusion-resistant signal for classification using simple neural networks, achieving accuracies above 0.9. They also demonstrate that seed data alone carry substantial discriminative power, and that generalised associahedra embedding explains part of the observed structure, with concrete integer ratios between seeds and quivers conjectured to hold at higher ranks. The results provide a data-driven lens on cluster algebra structure with potential implications for automating classification, exploring embeddings, and guiding future algebraic investigations, while offering publicly available code and datasets for reproducibility.

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.

Figures (10)

• Figure 2.1: The quiver for the $A_2$ example cluster algebra (a), as well as its exchange graph (b) where permutation equivalence is not applied. The respective clusters for each vertex in the exchange graph are: $\{ 0: [x_1,x_2], \ 1: [(x_2 + 1)/x_1, x_2], \ 2: [x_1, (x_1 + 1)/x_2], \ 3: [(x_2 + 1)/x_1, (x_1 + x_2 + 1)/(x_1x_2)], \ 4: [(x_1 + x_2 + 1)/(x_1x_2), (x_1 + 1)/x_2], \ 5: [(x_1 + 1)/x_2, (x_1 + x_2 + 1)/(x_1x_2)], \ 6: [(x_1 + x_2 + 1)/(x_1x_2), (x_2 + 1)/x_1], \ 7: [(x_1 + 1)/x_2, x_1], \ 8: [x_2, (x_2 + 1)/x_1, \ 9: [x_2, x_1] \}$, and the vertex mutated on to connect each seed is given as the respective edge feature. Note that throughout this paper vertices are labelled in order of generation, here this $A_2$ seed exchange graph is drawn with vertices in order of labelling, but may be be unfolded to show its single cycle nature with a different vertex ordering.
• Figure 3.1: Quivers defining the exchange matrices for the initial seeds. They are all rank 4, and generate cluster algebras of finite type (a), (b), (c); finite-mutation type that are not finite type (d), (e); and infinite type (f), (g). Vertices are labelled with respect to the row/column number in the exchange matrix; the double edge multiplicity in F4 indicates it is not skew-symmetric.
• Figure 3.2: The seed exchange graphs generated to depth 4 for each of the considered cluster algebras. Types are labelled, where finite-mutation are specifically not finite type so are infinite for these seed exchange graphs but finite for the respective quiver exchange graphs (not shown). Vertices are labelled in the order they are generated starting from the initial seed '0'.
• Figure 3.3: The number of seeds in the seed exchange graphs as depth varies for each of the considered cluster algebras, labelled by their respective initial seeds. Each type is depicted with a different linestyle.
• Figure 3.4: The size of the minimum cycle basis for each of the considered cluster algebras' seed exchange graphs as depth varies is plotted directly in (a), and the cycle basis length relative to the number of seeds is plotted in (b). Moreover, (c) focuses on the finite type A4 and D4 algebras with seed exchange graphs generated to their maximum depths; beyond depth 4 10-cycles are introduced so each cycle length frequency is denoted separately, as well as the total frequencies.