README.md

---
license: cc-by-2.0
pretty_name: Mutation equivalence of quivers
---

# Mutation Equivalence of Quivers

Quivers and quiver mutations are central to the combinatorial study of cluster algebras, 
algebraic structures with connections to Poisson Geometry, string theory, and 
Teichmuller theory. Quivers are directed (multi)graphs, and a quiver mutation 
is a local transformation centered at a chosen node of the graph that involves 
adding, deleting, and reversing the orientation of specific edges based on 
a set of combinatorial rules. A fundamental open problem in this area is 
finding an algorithm that determines whether two quivers are mutation equivalent 
(one can traverse from one quiver to another by applying mutations). Currently, 
such algorithms only exist for special cases, including types \\(A\\) 
[1], \\(D\\) [2], and \\(\tilde{B}\\), 
\\(\tilde{C}\\), and \\(\tilde{D}\\) [3]. The \\(\tilde{B}\\) 
and \\(\tilde{C}\\) types correspond to the classes \\(BD\\) and \\(BB\\) in 
our dataset, respectively. Consistent with Sage we use the naive notation, 
which specifies a quiver by indicating the two ends of the diagram, which 
are joined by a path [7]. To our knowledge, the remaining 
classes in this dataset ( \\(E\\), \\(DE\\), \\(BE\\)) lack characterizations. 

Recent work has explored whether deep learning models can learn to correctly 
predict if two quivers are mutation equivalent [4]. 
[5] utilized an alternative version of this dataset to re-discover 
known characterization theorems. The dataset consists of adjacency matrices for 
quivers drawn from 7 different mutation equivalence classes ( \\(A\\), \\(D\\), 
\\(E\\), \\(DE\\), \\(BE\\), \\(BD\\), and \\(BB\\)). 

## Dataset 

The task associated with this dataset involves identifying whether two quivers are mutation equivalent. 
Thus, the inputs are quivers (directed multigraphs). We chose to use examples with \\(11\\) nodes 
(though one could reasonably have chosen another number). They are encoded by their 
\\(11 \times 11\\) adjacency matrices and the labels are one of \\(7\\) different equivalence classes: 
\\(A_{11},BB_{11},BD_{11},BE_{11},D_{11},DE_{11},E_{11}\\). For the quiver mutation classes that 
are not mutation finite (that is, the mutation equivalence class has an infinite number of elements), 
the datasets contain quivers generated up to a certain depth, which is the distance from the original quiver, 
measured by number of mutations. The depths for those classes which are infinite are listed below 
and were chosen to balance the sizes of different classes.  

| Mutation equivalance class | Sampling depth |
|---|---|
| \\(BB_{11}\\) | 10 |
| \\(BD_{11}\\) | 9 |
| \\(BE_{11}\\) | 8 |
| \\(DE_{11}\\) | 9 |
| \\(E_{11}\\) | 9 |

Dataset statistics are as follows:

| | \\(A_{11}\\) | \\(BB_{11}\\) | \\(BD_{11}\\) | \\(BE_{11}\\) | \\(D_{11}\\) | \\(DE_{11}\\) | \\(E_{11}\\) | Total |
|---|---|--|---|---|---|----|----|---|
| Training | 11,940 | 27,410 | 23,651 | 22,615 | 25,653 | 23,528 | 28,998 | 163,795 |
| Test | 2,984 | 6,852 | 5,912 | 5,653 | 6,413 | 5,881 | 7,249 | 40,944 |


## Data generation

All mutation classes were generated using Sage [6], and the script can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/quiver_mutation_equivalence).

## Task

**Math question:** Find simple rules to determine whether or not a quiver belongs to a 
specific mutation equivalence class (out of classes \\(A_{11},BB_{11},BD_{11},BE_{11},
D_{11},DE_{11},E_{11}\\)). Note that rules for \\(A_{11}\\) and \\(D_{11}\\) are known.

**ML task:** Train a model that can predict a quiver's mutation equivalence class out of the 7 options above.

See the work [\[5\]](https://arxiv.org/abs/2411.07467) for an example of how a model 
trained on a variant of this dataset was used to re-discover known theorems.

## Small model performance

|  | Accuracy | 
|----------|----------|
| Logistic regression | \\(40.3\%\\) |
| MLP | \\(86.5\% \pm 1.9\%\\) | 
| Transformer | \\(92.9\% \pm 0.5\%\\) |
| Guessing largest class | \\(17.7\%\\) |

The \\(\pm\\) signs indicate 95% confidence intervals from random weight initialization and training.

## Further information

- **Curated by:** Helen Jenne
- **Funded by:** Pacific Northwest National Laboratory
- **Language(s) (NLP):** NA
- **License:** CC-by-2.0

### Dataset Sources

The dataset was generated using [SageMath](https://www.sagemath.org/). Data generation scripts can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function).

- **Repository:** [ACD Repo](https://github.com/pnnl/ML4AlgComb/tree/master/mheight_function)
  
## Citation

**BibTeX:**

    @article{chau2025machine,
        title={Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics},
        author={Chau, Herman and Jenne, Helen and Brown, Davis and He, Jesse and Raugas, Mark and Billey, Sara and Kvinge, Henry},
        journal={arXiv preprint arXiv:2503.06366},
        year={2025}
    }

**APA:**

Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (2025). Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics. arXiv preprint arXiv:2503.06366.

## Dataset Card Contact

Henry Kvinge, acdbenchdataset@gmail.com

[1] Buan, Aslak Bakke, and Dagfinn F. Vatne. "Derived equivalence classification for cluster-tilted algebras of type \\(A_n\\)." Journal of Algebra 319.7 (2008): 2723-2738.  
[2] Vatne, Dagfinn F. "The mutation class of \\(D_n\\) quivers." Communications in Algebra 38.3 (2010): 1137-1146.  
[3] Henrich, Thilo. "Mutation classes of diagrams via infinite graphs." Mathematische Nachrichten 284.17‐18 (2011): 2184-2205.  
[4] Bao, Jiakang, et al. "Machine learning algebraic geometry for physics." arXiv preprint arXiv:2204.10334 (2022).  
[5] He, Jesse, et al. "Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes." arXiv preprint arXiv:2411.07467 (2024).  
[6] Stein, William. "Sage: Open source mathematical software." (2008). 
[7] Musiker, Gregg, and Christian Stump. "A compendium on the cluster algebra and quiver package in Sage." arXiv preprint arXiv:1102.4844 (2011).