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README.md

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-license: cc-by-2.0
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+---
+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{D}\\) [3]). To our knowledge, the remaining classes in this dataset 
+(\\(E\\), \\(DE\\), \\(BE\\), and \\(B$\\) lack characterizations. Recent work has explored 
+whether deep learning models can learn to correctly predict if two quivers are mutation equivalent 
+[4]. [5] utilized a subset of this dataset to re-discover known characterization theorems. 
+
+## 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 |
+|---|---|
+| \\(B_{11}\\) | 10 |
+| \\(BD_{11}\\) | 9 |
+| \\(BE_{11}\\) | 8 |
+| \\(DE_{11}\\) | 9 |
+| \\(E_{11}\\) | 9 |
+
+Dataset statistics are as follows:
+
+| | \\(A_{11}\\) | \\(B_{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 [\[2\]](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.
+
+- **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).