Update README.md

README.md

@@ -6,17 +6,28 @@ 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. 
+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 
 
@@ -32,7 +43,7 @@ and were chosen to balance the sizes of different classes.
 
 | Mutation equivalance class | Sampling depth |
 |---|---|
-| \\(B_{11}\\) | 10 |
+| \\(BB_{11}\\) | 10 |
 | \\(BD_{11}\\) | 9 |
 | \\(BE_{11}\\) | 8 |
 | \\(DE_{11}\\) | 9 |
@@ -40,7 +51,7 @@ and were chosen to balance the sizes of different classes.
 
 Dataset statistics are as follows:
 
-| | \\(A_{11}\\) | \\(B_{11}\\) | \\(BD_{11}\\) | \\(BE_{11}\\) | \\(D_{11}\\) | \\(DE_{11}\\) | \\(E_{11}\\) | Total |
+| | \\(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 |
@@ -58,7 +69,7 @@ D_{11},DE_{11},E_{11}\\)). Note that rules for \\(A_{11}\\) and \\(D_{11}\\) are
 
 **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 
+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
@@ -104,9 +115,10 @@ Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (20
 
 Henry Kvinge, [email protected]
 
-\[1\] 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).  
+[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).