Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Authors: Jesse He
Helen Jenne
Herman Chau
Davis Brown
Mark Raugas
Sara C. Billey
Henry Kvinge
Venue: ICML 2025 poster (2025)
Keywords: AI for Math
Interpretability
Graph neural networks
Algorithmic reasoning
Primary area: applications->everything_else
TL;DR: We train graph neural networks to solve a problem arising in abstract algebra and then use interpretability techniques to independently discover a theorem

Abstract: Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate quiver mutation---an operation that transforms one quiver (or directed multigraph) into another---which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? In this paper, we use graph neural networks and AI explainability techniques to independently discover mutation equivalence criteria for quivers of type $\tilde{D}$. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D$, adding to the growing evidence that modern machine learning models are capable of learning abstract and general rules from mathematical data.

Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Jesse He Helen Jenne? 2 Herman Chau
Davis Brown? 2 Mark Raugas? Sara Billey3 Henry Kvinge?
Abstract
important role in mathematics research, modern machine learning tools like deep neural networks have only recently begun to be more broadly applied. From the other direction, increasing attention in machine learning has been given to whether complex models like neural networks can learn to reason when faced with mathematical or algorithmic tasks. For example, the phenomenon of algorithmic alignment in graph neural networks is known to improve the sample com- plexity of such networks when compared to networks with similar expressive power (Dudzik & Veličković, 2022; Xu et al., 2020), and graph neural networks remain competitive with transformers for recognizing local subgraph structures (Sanford et al., 2024).
Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of ex- amples so vast that they would be impossible for a single researcher to feasibly review and ana- lyze. In this work, we use graph neural networks to investigate quiver mutation-an operation that transforms one quiver (or directed multigraph) into another-which is central to the theory of cluster algebras with deep connections to geome- try, topology, and physics. In the study of cluster algebras, the question of mutation equivalence is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through sequence of mutations? In this paper, use graph neural networks and AI explainability techniques to inde- pendently discover mutation equivalence criteria for quivers of type D. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden rep- resentation that allows us to reconstruct known eri- teria from type D, adding to the growing evidence that modern machine learning models are capable of learning abstract and parsimonious rules from mathematical data.
Of course, working mathematicians often need more than just model that achieves high accuracy a given task. In many cases, a model is only useful if it learns features that can provide insight to mathematician. Further, one must be able to extract this insight from the model. In this work, we consider the specific problem of characterizing quiver mutation equivalence classes. We show first that in this setting a graph neural network learns representations that align with known non-trivial mathematical theory. We then show how one can use performant model to generate concise conjecture, which we then prove.
Introduced by Fomin and Zelevinsky in (Fomin & Zelevin- sky, 2002), quiver mutation is a combinatorial operation on quivers (directed multigraphs) which arises from the notion of a cluster algebra, an algebraic construction with deep con- nections to geometry and physics. Quiver mutation defines an equivalence relation on quivers (that is, it partitions the set of quivers into disjoint subsets). Two quivers belong to the same equivalence class if we can apply some appropri- ate sequence of quiver mutations to the first and obtain the second. Identifying whether such a sequence of mutations exists is generally a hard problem (Soukup, 2023). In some cases, however, a concise characterization which involves checking some simple conditions is known. For example, Theorem 5.1 (Buan & Vatne, 2008) and Theorem 5.3 (Vatne, 2010) tell us that we can check whether a quiver belongs to types A or D by verifying certain conditions relating to the presence of specific structural motifs. Henrich (2011) pro- vides a complete description of Dynkin and affine Dynkin type quivers in terms of certain families of infinite graphs.
1. Introduction
Examples play fundamental role in the mathematical re- search workflow. Exploration of a large number of exam- ples builds intuition, supports or disproves conjectures, and points the way towards patterns that are later formalized as theorems. While computer-aider simulation has long played
'Hllcooglu Data Science Institute, University of California San Diego San Diego, CA, USA Paacific Northwest National Labora- tory, Richland, WA, USA 'Department of Mathematics, University of Washington, Seattle, WA, USA. Correspondence to: Jesse He cjch020@ucsd.cdoo Henry Kvinge <henry.kvinge@ povl.gov>. gov>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
We train graph neural network (GNN) on a dataset consist- ing of 70, 000 quivers labeled with one of six different types (A. D, E, A, D, E). We find that not only does the re- sulting model achieve high accuracy, it also extracts features from type D quivers that align with the characterization from (Vatne, 2010). We identify the latter through a careful application of model explainability tools and exploration of hidden activations. Pushing this further, we carefully probe the hidden representations of type D, independently achiev- ing similar characterization' We describe how insights gained through the clustering of hidden activations and other explainability tools helped us prove our characterization of type D quivers (Theorem 6.1). This provides yet another example of how machine learning can be a valuable tool for the research mathematician.
quivers by computing all minimal quivers of infinite type. Since then, several other researchers have provided explicit characterizations of particular mutation classes of quivers (Bastian, 2011; Buan & Vatne, 2008; Vatne, 2010). Our main result follows these: we give an explicit characteri- zation of quivers of type D.,. akin to the characterization of quivers of type Dn, given in ( (Vatne, 2010). This differs subtly from (Henrich, 201 1), which characterizes finite-type quivers subgraphs of certain families of infinite graphs.
2.2. Mathematics and Machine Learning
Machine learning has recently gained traction as tool for mathematical research, Mathematicians have leveraged its ability to, among other things, identify patterns in large datasets. These emerging applications have included some within the field of cluster algebrus (Cheung et al., 2023; Bao et al., 2020; Dechant et al., 2023). Unlike our work, this research does not aim to establish new theorems around mutation equivalence classes, focusing rather on the per- formance of models on different versions of this problem. Armstrong- Williams et al. (2025) obtain result concerning the mulution-acyen -nccution-ucyllicityy of quivers, though their theoretical result guides their ML investigation rather than the reverse. Though unrelated to cluster algebras, Davies et al. (Davies et al., 2021) take an approach similar to the one taken here: using machine learning to guide mathematicians' intuition. They focus on two questions: one related to knot theory and one related to representation theory.
In summary, this work's 's contributions include the following: (1) We describe an application of graph neural networks to the problem of characterizing mutation equivalence classes of quivers. (2) Using Al explainability techniques, we pro- vide strong evidence that our model learned features which align with human- -eean-deveooped characterizations of quivers of type/ D from the mathematical literature. (3) Using insights gained from interpreting our model, we independently con- jecture and then prove characterization of quivers of type D in terms of certain subquivers.
2. Background and Related Work
Quivers and quiver mutations are central in the combina- torial study of cluster algebras, a relatively new but active research area with connections diverse areas of mathe- matics. For high-level discussion of quiver mutation in the broader context of cluster algebras, see Appendix A.1. Since quivers and quiver mutation can be studied indepen- dently of their algebraic origin, we have written the rest of the paper so that does not depend on this background.
Due to the existence of unambiguous ground truth and known algorithmic solutions, there has also been renewed interest in using mathematical tasks to better analyze how machine learning models learn tasks at a mechanistic level, including the emergence of reasoning in large models. For example, in (Chughtai et al., 2023), the authors use group operations to investigate the question of univer- sality in neural networks. Group multiplication is also used in (Stander et al., 2024) to investigate the grokking phenomenon. The idea of mechanistic interpretability- explaining model behavior by identifying the role of small collections of neurons is also demonstrated in (Zhong et al., 2023), where Zhong et al. are able recover two distinct algorithms from networks trained to perform modu- lar arithmetic, and (Liu et al., 2023), where Liu et al. find evidence that a network trained to predict the product of two permutations learns thoupeth structure.
2.1. Mutation-Finntn Quivers
In (Fomin & Zelevinsky. 2003), Fomin and Zelevinsky gave a complete classification of finite cluster algebras, which can be generated by a finite number of variables. (Equiv- alently, their associated quiver mutation classes are finite). Amazingly, they correspond exactly to the Cartan-Killing classification of semisimple Lie algebras. Their result says that a quiver associated to a cluster algebra of finite type must be mutation equivalent to an orientation of a Dynkin di- agram (Figure for examples). However, this result does not give an algorithm for checking this. To answer this question, Seven (Seven, 2007) gave a full description of the associated
3. Preliminaries
3.1. Quivers and Quiver Mutation
Aftter initially circulating our completed manuseript within the community, we were alerted that (Henrich, 2011) had already proved the theorem we discovered in somewhat different language.
In their work on cluster algebras (Fomin & Zelevinsky, 2002), Fomin and Zelevinsky introduce the notion of matrix mutation on skew-symmetric (or skew-symmettriablee) in-
2

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
leger matrices. By regarding skew-symmetric matrices as directed graphs, obtain combinatorial interpretation of matrix mutation in terms of quivers which are the central objects of our study. In this section we briefly describe preliminaries concerning quivers and quiver mutations.
Ex
2
3
4
5
a
12
Figure 2. Coxeter-Dynkin diagram for En, TE 2 6. Quivers of type En are only nutationnfinite for 6, 7,,,8 8,9.
Definition 3.1. quiver Q is directed (multi)graph with no loops or 2-cycles. There may be multiple parallel edges between vertices, represented as positive integer weights. The underlying graph of Q is the undirected graph where we ignore edge orientations.
The machine learning task: Train a classifier T to predict the mutation class of a quiver of type A, D, F,, Á, D, or E (Figure 3). We train & on a set of quivers with 6, 7, 8, 9, or 10 nodes, and test on quivers with 11 nodes. More details are provided in Section 4.1.
Definition 3.2. The mutatioo of a quiver Q at vertex j is the quiver (Q) obtained by performing the following: (i) For each path i + j + k in Q, add an arrow + k; (ii) Reverse all arrows incident to j: (iii) Remove any resulting 2-cycles created from the previous two steps.
As we will show through our explainability studies, a model trained on this task can learn rich features that point towards concise characterizations of these quivers.
Quiver mutation is an involution. That is, for vertex j in a quiver Q, My (1,(((Q)) = Q. Consequently, mutation is an equivalence relation on quivers.
3.2. Graph Neural Networks
Because quivers are represented as directed graphs, it is nat- ural to use graph neural networks to classify them. Graph neural networks (GNNs), introduced in (Defferrard et al., 2016; Kipf & Welling, 2017), are a class of neural net- works which operate on graph-structured data via a message- passing scheme. Given (attributed) graph G EV) with node features E R for each node E V and Cuv E R for each edge (u, v) E E, each layer of the net- work updates the node feature by aggregating the features of its neighbors. The final graph representation is computed by pooling the node representations. Because prior work has characterized quiver mutation classes based the presence of particular subgraphs, we use the most expressive GNN architecture for recognizing subgraphs (Xu et al., 2019). To this end, we adopt version of the graph isomorphism network (GIN) introduced in (Xu et al., 2019) and modified in (Hu et al., 2020) to support edge features. Since quivers are directed graphs, we adopt a directed message-passinn scheme with separate message-passing functions along each orientation of edge. We refer to architecture Directed Graph Isomorphism Network with Edge features (DirGINE), and denote the network itself by 4. We describe our DirGINE architecture in greater detail in Appendix B.
My
Figure 1. An example of a quiver and mutation at vertex j.
Definition 3.3. We say two quivers Q, Q' are mutation equivalent if Q' can be obtained from Q (up to isomorphism) by sequence of mutations. We refer to the mutation class of Q as the set [Q] of (isomorphism classes of) quivers which are mutation-equee -eaaaton-euuivllnnnt equivallntiiiiiieeee to Q.
Definition 3.4. We say a quiver Q is mutation-finite if its mutation class [Q] is finite, and mutation-infimite otherwise.
Definition 3.5. Given starting quiver Q, the mutation depth of a quiver Q' E [Q] (with respect to Q) is the mini- mum number of mutations required to obtain Q' from Q-
We will consider the mutation classes of quivers that are of simply laced (that is, with no parallel edges) Dynkin or extended (affine) Dynkin type. These are the quivers whose underlying undirected graphs are shown in Figure 3. In particular, the quivers of type D and type D will be the focus of our explainability analysis. We also include quivers of type F (Figure 2) which are not finite or affine type in our training and test sets. (The diagram En is only finite type for 6, 7, 8, and affine for n = =9)
4. Methods
4.1. Model Training
We train a 4-yyyer DirGINE GNN with hidden layer width of 32 to classify quivers into types A, D, E, A, D, and F. The training data consists of quivers of each type on 6. 7, 8, 9, and 10 nodes. The test set consists of quivers of types A, D, E, Å, on 11 nodes. (Type E is not de- fined on 11 nodes.) We generate data with Sage (The Sage Developers, 2023; Musiker & Stump, 201 1), which we de-
The following well-known lemma (Vatne, 2010) ensures the mutation classes of the simply laced Dynkin diagrams are well-defined.
Lemma 3.6. If quivers Qı and Q2 have the same underly- ing graph T and T is a tree, then Q1 and Q2 are mutation equivalent.
3

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
An
Da
Da-1
Ba
Fq
Bg
Fi
Rs
Figure 3, Simply laced Dynkin diagrams and their extensions.
Loss
Accuracy
4.2. Size Generalization
1.0
Train Test
1.0
By training quivers of sizes 7-10 and testing quiv- ers of size 11, we encourage our DirGINE to learn size- generalizable features, leveraging the promise of size gener- alization in GNNs. Given this, it is natural to ask how well our model performs for larger n. (After all, any classifica- tion rules for quivers should be size-gence size-generalizable as well.) To examine this question, we test our model on quivers of types A, D, E A, and D on Because the number of distinct quivers grows quickly with size, we only use subsample of each class, which we dis- cuss in greater detail in Appendix C. The results (Table 1) indicate that our GNN generalizes well, albeit not perfectly. We believe this is because our GNN has a fixed depth of 4, and hence cannot recognize the larger substructures that may appear in larger quivers. In particular, we will in Section 5.2 that some quivers can be distinguished by long cycles, which message- message-passing networks may fail to recognize (Chen et al., 2020).
0.8
0.9
0.6
0.7
0.6
Train
0.5
12, 13, ..... 20 vertices.
20
40
20
40
Epoch
Epach
Figure 4. Average cross-entropy loss (left) and classification accu- racy (right) on train and test sets across 10 trials of training. Testing accuracy is consistently higher than training accuracy, perhaps due to the absence of class E in the test set and the fact that Es- Eg in the train set.
scribe in greater detail in Appendix C. Figure shows the average cross-entropy loss and classification accuracy by epoch across 10 trials. We take the best epoch from training (99.2% test accuracy) for our analysis.
4.3. Explaining GNNs
In order to extract mathematical insight from a trained GNN model ④ we require a way to explain its predictions by identifying the substructures that are responsible for its pre- dictions. That is, for each graph G, we wish to identify a small subgraph Gs such that (GG = (Gs). We use the GNN explanation method PGExplainer (Luo al., 2020), which trains neural network graphs. For an input graph G, the explanation network produces an attribution Wa,e for each edge (u, v) using the final representations for nodes and v.
While the differences between the train and test set (par- ticularly the absence of type from the test set) might be problematic if our goal was to assess whether a machine learning model can classify quivers into mutation types, our primary goal to extract mathematical insights from the features the model learns for types D and D. As such, we use the test set to indicate whether a model was sufficiently performant to justify the application of cxplainability tools. Moreover, the inclusion of types E and E help modulate the difficulty of the classification problem, ensuring that the model learns discriminative features for classes and Đ.
identify important sub-
While PGExplainer's effectiveness mixed across differ- ent comparisons (Agarwal al., 2023; Amara et al., 2022).

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
π
12
13
14
15
16
17
18
19
20
Accuracy
99.6
98.7
97.7
95.5
94.3
92.0
91.1
89.4
89.1
Table 1. Accuracy of our trained DirGINE on unseen quivers of size n 12, 13, .,,,, 20.
it is effective at providing model-level substructure expla- nations for graph classification tasks. For example, when applicd to a GNN trained on the MUTAG dataset (Debnath et al., 1991) to predict the mutagenicity of molecules, PGEx- plainer is regularly able to identify that the model predicts mutagenicity based on the presence of nitro (NO2) groups (Luo et al., 2020). As will see in Section 5, the ability of PGExplainer to identify explanatory graph motifs makes it suitable for our purpsss.. To analyze our trained Dir- GINE, we train PGExplainer on 1000 randomly selected instances from the train set for epochs. Further discussion of PGExplainer provided in Appendix B.2.
5.2. The Mutation Class of D,, Quivers
We now describe the classification of the mutation class M of Type D, quivers by Vatne (2010). As with type A, we use MM to denote the collection of type D quivers for all n 2 3 (noting that A3 = D3 and that D1, D2 are not defined):
R
(2)
In Vatne's classification, each type is collection of subquiv- ers joined by gluing vertices. The proof relies on the fact that if the vertex joining the blocks is a connecting vertex (defined below), one can mutate a sub-quiver of type A to the quiver in (1) without ever mutating at the connecting vertex. Formally, we have the following.
5. Extracting Quiver Characterizations
Before we state Theorem 6.1, which we re-discover through analysis of our GNN model, we describe the known char- acterizations of mutation class A quivers (Buan & Vatne, 2008) in Section 5.1 and mutation class quivers (Vatne, 2010) in Section 5.2. We also describe in Section 5.2 how reconstruct the characterization of type D (Vatne, 2010) by probing our trained GNN?.
Definition 5.2. For quiver I E .M we say vertex of T is a connecting vertex if e is either degree one or degree two and part of an oriented 3-cycle.
Vatne proves the following classification of quivers of M into four types by first proving that each subtype is mutation- equivalent to (2), then proving that the collection of quivers described by these subtypes is closed under quiver mutation.
5.1. The Mutation Class of A,, Quivers
The class of An quivers consists of all quivers which are
mutation equivalent to (I).
Theorem 5.3 (Vatne 2010). The quivers of the mutation class of Dn consist of four subtypes shown in the diagrams below. In these diagrams, r, I', and I" are full subquivers of mutation class A with connecting vertex. Unoriented edges mean that the orientation does not matter.
The following theorem provides a combinatorial charac- terization of all such quivers. We refer to the collec- tion of all such quivers as MA We will also denote M = M
Type
P
(3)
Theorem 5.1 (Buan & Vatne 2008). A quiver Q is in the mutation class MÅ if and only if: (i) All cycles are oriented 3-cycles. (ii) Every vertex has degree at most four. (ii)) Ifa vertex has degree four, two of its edges belong the same 3-cycle, and the other two belong to different 3-cycle. (iv) If vertex has degree three, two of its edges belong to a 3-cycle, and the third edge does not belong to any 3-cycle.
Type II
(4)
r
r
While this result is interesting in its own right, the impor- tance for this paper is that the quivers in the mutation class of type Dn contain subquivers of type A.
Type III.
(5)
Unlike Theorem 6.1, were already aware of the type A and type characterizations when we began this work.
r
r
5

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Type IV.
sis, combined with the PGExplainer attributions, provides strong evidence that a GNN is capable of re-discovering the same abstract, general characterization rules that align with known theory through training on a naive classification task.
r
F
1"
5.2.1. Do CHANGES IN SUBTYPE MOTIFS ACTUALLY IMPACT PREDICTIONS?
d
(6)
To further understand how the model is using the type D- specific subquivers from Theorem 5.3 in practice, we ex- amine the model's predictions when the edges identified by PGExplainer are removed. If the model is primarily keying into the type D motif, removing this should result in the quiver being predicted as type A.
In particular, we draw attention to the oriented central cycle in type IV. Every edge in the central cycle may be part of an oriented triangle with a connecting vertex not on the central cycle (called a spike).
We find that across all 32, 066 test examples from type D, a plurality (14,916 or 46.5%) of the predictions flip to A, as we would expect if it was using the characterization from Theorem 5.3. Of the remaining examples, most (14,238 or 14.4% of the total) flip to a predicted class of E, with the next-largest being D (no flip) at 2,581 or 8.0%. Finally, 264 quivers (0.08%) flip to Á, while 67 quivers (0.02%) flip to D. None of the predictions flip to É.
Because our DirGINE can, in general, learn to recognize certain subgraphs in a quiver, it is reasonable to ask whether the model was relying on the same subtype motifs identi- fied by human mathematicians. We investigated the case of the type D classification from Theorem 5.3 using PGEx- plainer as described in Section 4.3, Some sample soft masks produced by PGExplainer are shown for quivers correctly classified as type in Figure 5. Dark red edges are judged more important for the type D prediction by PGExplainer, while lighter edges are judged less important. In each case, the explanation highlights edges which align with Theo- rem 5.3. Figure 5 contains an example of each of the four types. The reader can check that in each case the edges of the relevant subtype motif tends to have substantially higher attribution. This initial analysis seems to suggest that our model has independently learned the same subtype motifs for classifying type D quivers from known (human) theory. The attributions shown in Figure 5 also suggest that the model relies on the opposite end of the quiver-a behavior that seems superfluous for recognizing type D. We will show in Section 6 that this is actually very important for distinguishing quivers of type D from D.
Why are there so many instances that flip to type E? This may be due to the PGExplainer attributions for type D being inexact, perhaps because PGExplainer generalizes imper- fectly or because some edges do not contribute positively to D but rather contribute negatively to other classes. As a re- sult, many of the quivers where we remove highly attributed edges may be out-of-distribution for the model. Since type E is the only class which contains mutation-infinii quivers, it is perhaps not surprising that the model would predict these out-of-distributio quivers are of type E.
6. Characterizing D Quivers
In this section, we describe how our trained model and ex- plainability techniques enable us to independently discover characterization of the mutation class of D quivers, stated in Theorem 6. below. We learned after initial circulation of this work that this result can be derived from (Henrich, 2011). However, as our characterization closely reflects the way in which we analyzed our model, we felt it would still be of interest to the machine learning community as an example of extracting research-level mathematics from a deep learning model. Due to the complexity of the charac- terization, some of the details of the characterization are left to Appendix D. and the proof to Appendix E.
Figure 5 strongly suggests that our GNN recognizes the same subtypes as in Theorem 5.3. However, one should be careful in this interpretation, as there is a substantial literature showing that it is easy to misinterpret post-hoc explainability methods (Ghorbani et al., 2019; Kindermans et al., 2019). Thus, we also examine the embeddings of type D, quivers in the model's latent space. We use prin- cipal component analysis (PCA) to reduce the dimension of the embedding from the model width of 32 to dimen- sions for visualization. The resulting graph embeddings, plotted in Figure 6, show a clear separation of the different subtypes. In fact, the layer 3 embeddings in the original 32-dimensional embedding space can be separated by a linear classifier with 99.7 + 0.0% accuracy. Subtypes through IV are not labeled in the training data, so this analy-
Theorem 6.1. The mutation class of class i quivers is MD 1. the collection of quivers of paired rypes together with Types V, Va, Vb, V", Va', Vb', VI, and VI'.
Similar to Vatne's classification of the mutation class of Dn quivers, our classification consists of different subtypes.
6

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Quiver 16153, Class: Prediction: D,
Quiver 301.2. D. Prediction:
Quiver 21573, Prediction: D.
Quiver 21175, cass: Predictions
Prediction:
Masked Prediction:
Prediction
1.0
1.0
1,0
0.
56
-0.
-32
02
52
022
Figure 5. Edge attributions from PGExplainer lype D11 quivers of each subtype. The masked prediction is the GNN prediction when highly attributed edges are removed. From left right: Type L. Five dark red edges highlight subquiver consisting the leaves 10, the connecting vertex 1, and the 3-cycle that the connecting vertex is part of; Type II. Five dark red edges highlight the center block seen the Type II diagram(the vertices where the vertex 6 is the vertex el; Type IV. The dark red edges highlight the oriented 5-cycle and the spike _ 10 4 +.
8 are and c', respectively); Type III. Four dark red edges highlight the oriented 4-cycle,
40
20-
5-
20
10
Type Type
Type II
Type IV
-5-
0-
-10
-10
-15
-30
20
-10
0
10
-10
100
20
30
40
PC1
PC
PC.1
Figure PCA latent space embeddings for mutation class D quivers colored by subtypes.
However, there are many more subtypes compared to the type D case, so we find convenient to organize them into families: what we call paired types, quivers with one central cycle, and quivers with central cycles.
can obtain any pair of subtypes of type D. Because the placement of e arbitrary, we see that many type ò.-1 quivers can be described by two of the type subtypes characterized in Section 5.2 which share type A piece Te. We will refer to such quivers Types 1-1, 1-II, I-III, etc., and collectively as paired types. (See Figure 13 in Appendix F for all paired types.) It remains, then, to identify the quivers in this mutation class which not are of paired type.
As in Types A and D, Lemma 3.6 means that we may choose an arbitrary orientation of the extended Dynkin diagram Dn-1- will be convenient to begin with the orientation (7), viewing as two quivers Q1 and Q2 of type (2) connected their roots by a connecting vertex c.
While a human mathematician could conceivably discover the same characterization of ò quivers simply by begin- ning with (7) and exhaustively performing mutations, the mutation class of î quivers admits many diverse subtypes compared to classes A or D. This increased complexity cre- ates some difficulty (and perhaps more importantly, tedium) in examining examples manually. By taking advantage of machine learning, we are able to quickly organize examples into distinct families examine.
12
(7)
From the orientation in (7) is immediately clear that by mutating Q) and Q2 independently without mutating e, we
Based strategy in Section 5.2, plot PCA reduc-

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Layer 1
Layer
Layer
15
00-
10
50-
40-
HI
20-
SIV
20
11-11
10
BV
-10
-10-
-30
-20
-30 -20
-10
20
-20
20
-20
20
PC
PC
PC
Figure PCA reductions of latent space embeddings mutation class Dao quivers colored by paired types or "Other".
tions of the latent space in Figure We can see that the quivers that do not correspond to paired subtypes, colored "Other", separate clearly into two clusters layer 3. By isolating these quivers and performing k- k-means clustering with k 2, the model guides our characterization of the remaining class D subtypes. Figure 8 shows examples from each cluster. (More examples are given in Appendix F.) The key insight we gain from examining the quivers in each cluster is that the remaining subtypes can be separated by the number of Type IV-like central cycles.
we show one example of Type (8). The rest the Type V family, types Va, Vb, V', Va", and Vb', can be obtained from (8), we describe in Appendix D.1.
The Type VI family consists of the type D quivers with two central cycles. An example of Type VI is provided in (9), where we color each central cycle for clarity, Type VT (10) is an exceptional version of type VI in which both central cycles are of length and is allowed be connecting vertex for a type quiver. Theorem 6.1, which we prove in Appendix E, states that these subtypes together with the paired types give exhaustive characterization of the mutation class of Dn-1 quivers.
I
b
Q
(8)
(10)
r
7. Conclusion
In this work we analyze a graph neural network trained to classify quivers as belonging to one of 6 different types, motivated by the theory of cluster algebras and the problem of quiver mutation equivalence. Using explainability tech- niques, provide evidence that the model learns prediction rules that align with existing theory for one of these types (type D). Moreover, the model behavior which allows us to recover this result emerges from the model an unsuper- vised manner-the the model is not given any subtype labels, and yet is able identify relevant blocks recognize type D quivers. Applying the same explainability techniques to another case, we also independently discover and prove characterization of the mutation class of D,-1 quivers. Taken together, our work provides more evidence support- ing the idea that machine learning can be a valuable tool in the mathematician's workflow by identifying novel patterns in mathematical data.
r
F
1
The type D quivers with a single central cycle make up the Type V family. These all have a sort of "double spike" motif;
8

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Quiver 134448, Class: D. Prediction:
Layer 3
Quiver 134919, Class: O Prediction:
1.0
00
0.8
0.5
Prodlccoo Imporaano
Prediced Importance
0.6
-0.5
20-
0.2
-02
0.2
0.0
0.0
Figure 8. PCA of clustered layer 3 latent space embeddings of "Other" quivers in Mie (middle), with selected examples from each cluster (left, right). Edges are colored by PGExplainer attributions. The quiver on the left is of Type V while the quiver on the right is of Type VI.
Impact Statement
Xiao, Y. Quiver mutations, seiberg duality, and machine learning. Physical Review D, 102(8):086013, 2020.
This work showcases an application of explainability ad- vance application in algebraic combinatorics. It shares the societal consequences of machine learning and alge- braic combinatorics in general, none which we feel must be specifically highlighted here.
Bastian, J. Mutation classes of An-quivers and derived equivalence classification of cluster tilted algebras of type An Algebru & Number Theory, 5(3):567-594, Decem- ber 2011. ISSN 1937-0652,
doi: 10.2140/ant 1.5.. 567. URL http: // http//xxxxooo///////oo/////.dd org/ /10. 2140/art 2011.5 567..
Acknowledgements
The authors would like thank Scott Neville and Kayla Wright for helpful discussions. We also thank Pavel Tu- markin for bringing to our attention the work of Henrich (2011), This research was supported by the Mathematics for Artificial Reasoning in Science (MARS) initiative via the Laboratory Directed Research and Development (LDRD) in- vestments at Pacific Northwest National Laboratory (PNNL). PNNL is multi-program national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-ACO5-76RLO--18 1830..
Beddar- Wiesing, S., D'Inverno, G. A., Graziani, C., Lachi, V. Moallemy-Oureh, A., and Scarselli, F. On the ex- tension of the Weisfeiler-Lchman hierarchy by WL tests for arbitrary graphs. In 18th International Workshop on Mining and Learning with Graphs, 2022. URL https: // /ppeneeview net/ forum? d- 6GrgDz2y5.
Beddar-Wiesing,
Buan, A. B. and Vatne, D. F. Derived equivalence classifi- cation for cluster-tiledd algebras of type An. Journal of Algebra, 319(7):2723-2733 2008
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
A. Additional Background
A.1. Cluster algebras and quiver mutations
A cluster algebra is a special type of commutative ring that is generated (in the algebraic sense) via a (possibly infinite) set of generators that are grouped into clusters. A cluster algebra may have finitely or infinitely many generators, but the size of each cluster is always finite and fixed. A cluster algebra is said to be of rank n if each of the clusters contains generators, called cluster variables. These clusters are related via exchange property which tells us how transform one cluster to another (Fomin& & Zelevinsky, 2002).² It turns out that there is nice combinatorial interpretation of this transformation when we interpret clusters as quivers with each generator corresponding to vertex in the quiver. Then quiver mutation describes this exchange of cluster variables. In this setting, the mutation equivalence problem asks when two clusters generate the same cluster algebra.
Quiver mutation also appears in physics in the form of Sieberg duality. Seiberg duality is an important low-energy identification of naively distinct non-Abelian four dimensional N = supersymmetric gauge theories, where gluons and quarks of one theory are mapped to non-Abelian magnetic monopoles of the other, and vice versa, but result non- trivially in the same long-distance (low-energy) physics (Seiberg, 1995). Quiver gauge theories in string theory are N = 1 supersymmetric quantum field theories that have field and matter field content defined in terms of their superpotential read off from an associated quiver diagram. They arise in a number of scenarios, including the low energy effective field theory associated to D-branes at a singularity. The superpotential and its associated quiver are defined by the representation theory of the finite group associated to the singularity (e.g., if is of ADE type), via the McKay correspondence (Greene et al., 1999). Associated to quivers are cluster algebras, and quiver mutations this physical context are maps that identify naively distinct N = 1 4d supersymmetric gauge theories under Sieberg duality in the low-energy (IR) limit. There is an extensive literature in this area of physics, some earlier work has used machine learning techniques to study pairs of candidate Seiterg dual physical theories related by quiver mutations (Bao et al., 2020). We hope that the present work may be helpful for those working to develop an improved mathematical understanding of both Seiberg duality and quiver gauge theory in general.
A.2. Mutation-Finite Quivers
Quivers of type D,,-th the main subject of our study are mutatioonfiintte That is, they have finite mutation equivalence class. Mutation-finite quivers and their associated cluster algebras are of interest to many cluster algebraists. Felikson, Shapiro, and Tumarkin 2012b gave a description of the mutation- finite quivers in terms of geometric type (those arising from triangulations of bordered surfaces), the Es, Er, Egg Dynkin diagrams and their extensions, and two additional exceptional types X6 and X identified by Derksen and Owen (Derksen & Owen, 2008). Specifically, they showed that mutation- finite quivers must either be decomposable into certain blocks or contain subquiver which is mutation equivalent to Eg or Xg. It will be occasionally useful to refer to these blocks in in Appendix D, so we describe them here. However, block decompositions are not unique in general, so we will not place too much emphasis on the block decompasitions of each quiver we describe.
Definition A.1 (Felikson et al. 2012a). A block is one of six graphs shown in Figure 9, where each vertex either an outler dead end (Fomin al., 2008). A connected quiver Q is dlcom-decomeesaale if it can be obtained by gluing together blocks at their outlets, such that each vertex is part of at most two blocks. Formally, one constructs blook-decomposable quiver as follows:
(i) Take partial matching of the combined set of outlets (no outlet may be matched to outlet from the same block);
iii Identify the outlets in each pair of the matching;
(iii) If the resulting quiver contains a pair of edges which form a 2-cycle, remove them.
It is worth noting that classification in the mutation-finit setting has proven to be more challenging than in the finite setting. The classification of mutation mutation-finite cluster algebras in the case with no frozen variables was achieved nearly a decade after Fomin and Zelevinsky classified finite cluster algebras (Felikson et al., 201 2012);a), and the general case was solved only last year (Felikson& & Tumarkin, 2024).
In general, cluster consists both cluster variables and generators known as frozen variables (that lack this exchange property).
12

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
0
B
B1
Bilta
B
BIv
By
Figure 9. Blocks of type I-V introduced by (Fomin et al., 2008). Open circles denote outlet,, which may be identified with at most one outlet from another block. Closed circles represent dead ends, which may not be identified with any other vertex.
B. Implementation Details
B.1. Model Architecture
Here we describe in detail our Directed Graph Isomorphism Network with Edge Features (DirGINE). The DirGINE uses a message-passing
message- passing scheme, maintaining a representation of each node. In each layer, each node's 's representation is updated according to a parameterized function its neighbors' representations. Formally, the f-th layer is given by
2 ReLU
Σ (N.DEE
Pin (x (xlff-1), Euv) ewo)+
(2)
Σ (NAB)E
Pout (ww
(o ever)
(11)
where (OO is affine transformation and Pinn and y are feedforward neural networks with 2 fully connected layers. Because we are classifying graphs, we use sum pooling. That is, in the final layer L we can assign vector to the entire graph G by adding the vectors associated with each vertex in the layer. We write
(GG) = (G)
Σ (1).
(12)
VEV(G)
The expressive power of graph neural networks is intimately connected to the classical Weisfeiler-Lcmmnn (WL) graph isomorphism test (Weisfeiler & Leman, 1968). Given an undirected graph with constant node features and no edge features, a graph neural network cannot distinguish two graphs which are indistinguishable by the WL test (Xu et al., 2019), and graph neural networks are able to count some (but not all) substructures (Chen et al., 2020). In our case, operating on directed graphs with edge features slightly enhances the expressive power our network, as the WL tests for attributed and directed graphs strictly stronger than the undirected WL test (Beddar-Wiesing Wiesing et al., 2022). As we saw in Section the ability to distinguish directed substructures is crucial their application in classifying quiver mutation classes. (For example, distinguishing Type D, from Type Ån-1 quiver.)
B.2. Further Discussion of PGExplainer
While number of post-hoc explanation methods exist for GNNs, most fall into of two categories:
cundient-based methods use the partial derivatives of the model output with respect to input features. A larger gradient is assumed to mean that a feature more important.
(ii) Perturbation--asel methods observe how the model's predictions change when features are removed or distorted. Larger changes indicate greater importance.
The method we adopt, PGExplainer (Luo al., 2020), is perturbation- method which produces edge attributions using neural network g. Using the final node embeddings of and v as well as any edge features euvs y produces an attribution
Way gx (L) ewv))
(13)
We use the implementation provided by PyTorch Geometric (Fey Lenssen, 2019), where is implemented as an MLP followed by a sigmoid function to ensure that S Whr < Then rather than produce a "hard" subgraph as our explanatory
13

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
graph Gg. the attribution matrix Q2 (wxy) can be seen "soft" mask for the adjaceney matrix A(G). That is, instead of providing binary 0-1 attribution for each edge, PGExplainer provides attribution waj E [0, 1]. We then use the weighted graph with adjacency matrix Q o A(G) for Gs (where Q o A(G) is the elementwise product),
PGExplainer follows prior work (Ying et al., 2019) in interpreting Gs as a random variable with expectation ? = E[A(Gs))
where each edge (i. is assigned a Bernoulli random variable with expectation wiy. PGExplainer then attempts to maximize the mutual information I (Q), Gs). However, because this is intractable in practice, the actual optimization objective is
min (E( (GGs)) + + + (H())
(14)
Here CE( (GG)) is the cross-entropy loss between the predictions ((G) and ((Gs), [[[]] is the Lı-norm of ?
( - Σ Σ [11 wwi)) log(1 wa) + Wij log(wss)]:
(15)
(..)EEL.
and a and I are hyperparameters. In our analysis, use hyperparameter values 2.5 and B 0.1. The |[2]]] term acts as a size constraint, penalizing the size of the selected Gs. The (Q2) term acts as connectivity constraint, penalizing instances where two incident edges are given very different attributions. By training a neural network to compute Ω PGExplainer allows us to generate explanations for new graphs very quickly, as well as take a more global view of the model behavior.
C. Data Generation and Model Training
Quivers were generated using Sage (The Sage Developers, 2023; Musiker & Stump, 2011). For training and inference, each quiver was converted to PyTorch Geometric (Fey & Lenssen, 2019). Following the representation convention in Sage, A parallel edges are represented by a single edge with edge attribute (ks -k), and each vertex is initialized with constant node feature. The train set consists of:
All quivers of types A, D, Å, and D on 8, 9, and 10 nodes.
All quivers of type É. (Type É is only defined for 7, and 9 nodes, corresponding to extended versions of Es. Er. and Eg, respectively. All quivers of type E are mutation-finies
All quivers of type E for 6, 7, 8. (The Dynkin diagram Eg is the same the extended diagram Eg.) Type E is only mutation-finite for 6, 7, 8. and coincides with Eg for = 9.
Quivers of type E10 up to mutation depth of 8, with respect to Sage's standard orientation for E10 (Figure 10). (While type E' is mutation finite for < 9, Eio is mutation-infinite)
3
4
6
7
9
10
Figure Default orientation of E1o Sage. Mutation depth is assessed with respect to this orientation for generating data in Sage.
The test set consists of quivers on 11 nodes. We use all quivers of type An1 Aιo, D11 and Dio. and again generate quivers up to mutation depth for E.,. The number of quivers of each size from each class can found Table Note that type É is absent from the test set, because E is not defined for 11 nodes. The class of type À quivers is also unique, as the collection of A is actually partitioned into [n/2] distinct mutation classes, described by (Bastian, 2011).
For the size generalization experiment in Section 4.2, we only generate quivers up to mutation depth of 6, as the number of distinct quivers grows too quickly with size to generate classes exhaustively. When the number of generated quivers exceeds 100,000, we randomply subsample 100,000 quivers to avoid out-of-memory errors.
We train with the Adam optimizer for 50 epochs with a batch size of 32 using cross-entropy loss with L₁ regularization (7 x 10-6) using an Nvidia RTX A2000 Laptop GPU.
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Train
An
Dn
En
An-1
Dn-1
En
7
150
246
416
340
146
132
442
810
1,574
1,265
504
1,080
1,424
2,704
4,582
1,868
4,376
10
4,522
9,252
10,906
16,382
6,864
Test
11
14,924
32,066
24,060
63,260
25,810
Table 2. Number of quivers of cach type and size in train and test sets.
D. The Mutation Class of D Quivers
Theorem 6.1 groups the mutation class of D quivers into three families; paired types, quivers with one central cycle, and quivers with two central cycles. A complete list of paired types is shown in Figure 13. Here we provide additional details about the subtypes with one and two central cycles.
D.1. One central cycle
Types V, Va, Vb. Type V quivers resemble Type IV quivers of class D, but one edge in the central cycle is part of Biv block that appears in the Type II quiver of class D. In the diagram below, this is the edge a: a - b. Note that no larger subquiver may be attached to and d'.
d
r
M
b
a
(16)
d'
Mutating Type V produces the subtype Type Va, which is similar, but the block Biy replaced with an oriented 4-cycle, shown below.
d
M
b C
(17)
d'
15

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Mutating type Va at d' produces the subtype Type Vb, which is similar to (16) except the block Biv is reversed.
d
r
F
b
(18)
d'
Mutating again d creates quiver isomorphic Type Va by swapping d and d', and finally mutating once more at d' results in Type V again. Notice that the choice of d and d' is arbitrary.
Types v,, Va', Vb'. The rest of this cluster consists of the following types, which we call V", Va", and Vb" as they are related to each other by an analogous sequence mutations. That is, starting from V', performing the sequence of mutations pa, pad, pa. par yields Types Va', Vb', Va', V", in that order. Moreover, He converts each Type V Va', Vb' into a corresponding Type V, Va, Vb quiver, respectively. The Type V Type V case is shown in Figure 11.
Type
Type Va'
Type Vb
d
d'
d'
d
d'
b
(19)
a
(20)
b (21)
r
r
r
To see that these types are mutation equivalent to (7), it suffices to show that Type V are mutation equivalent to one of the paired types.
Lemma D.1. Type V quivers (16) are mutation equivalent to (7).
Proof. We mutate at vertex a. There are several cases. Recall from Theorem 5.3 that a spike refers to oriented triangle on the central cycle.
the central cycle is of length > 3, there are two subcases:
If there is spike at vertex c, then the resulting quiver is of Type II-IV, where the vertices e and a are playing the roles of and c" in Figure 13, respectively, and b plays the role of c".
(b) Otherwise, the resulting quiver is of Type I-IV, where d and d' are the pair of dead ends.
If the central cycle is length 3, say triangle + b + + then there are four subcases, depending on the presence of spikes on the central cycle:
If the central cycle has no additional spikes, then μa yields a Type I-I quiver.
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
(b) If is part of spike but is not, then the result is Type I-Il quiver, where b and v the pair of dead ends on the Type side and d, d' the dead ends in the Biy block in the Type II side.
If is not part of spike but is then the result is Type I-III, where and d' the pair of dead ends in the Type I side.
If both a and are parts of spikes, then the result is Type II-III with dead ends d, d' in the Biv block the Type side.
Corollary D.2. Quivers of Types Va (17) and Vb (18) are mutation equivalent to (7).
Corollary D.3. Type V quivers (19) are mutation equivalent (7).
Corollary D.4. Quivers of Types Va" (20) and Vb' (21) mutation equivalent (7).
d'
d'
Pe
b
b
Figure 11. Performing Me convert between quiver of Type V' (left) and Type V (right). Note that in the Type quiver, the central cycle is + c + + o, so the vertex e is not a connecting vertex as it is in 16.
This completes the description of new types with one central cycle.
D.2. Two central cycles
The other family, with quivers with two central cycles, consists of the following:
Type VI. Quivers Type VI consist of two Type IV quivers which share one vertex c among both central cycles, and are further joined by two edges that create oriented triangles for which e is a vertex. These oriented triangles can be seen as shared spikes among both central cycles. In (22), we color one central cycle blue and one red for clarity. The central cycles
17

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
may be any length 2
F
(22)
P"
Type VT' In Type VI, the shared connecting vertex e allowed be a connecting vertex for a larger subquiver of Type A in general. However, there is exception to this when both central cycles triangles and have no additional spikes. The result is block By whose outlet is connecting vertex for type A subquiver r. We refer to this Type VT'. Notice that if the quiver has only five vertices, then is only one vertex, in which case there is no difference between Types VI and Vr.
T
(23)
Again, to show that these are mutation equivalent to (7), suffices to reduce the paired types.
Lemma D.5. Type quivers (22) are mutation equivalent to (7).
Proof. If both central cycles are of length > 3, then mutating at vertex results in a quiver Type IV-IV. If a central cycle is of length 3, then He turns that central cycle into a subquiver of Type or Type III, depending on whether or not that central cycle does not have or does have a third spike, respectively.
For the case of Type VI' quivers, the following lemma from (Vatne, 2010) will be useful:
Lemma D.6 (Vatne 2010). Ler T E MA n 2 2, and let e be connecting vertex for Т. Then there exists sequence of mutations on I' such that:
(i) Pe does not appear in the sequence (that is, do not mutate at c);
(ii) The resulting quiver is isomorphic to (1) in Section 5.1;
(iii) Under this isomorphism, is mapped to 1.
Lemma D.7. Type VI' quivers (23) mutation equivalent (7).
Proof. By Lemma D.6, we may mutate so that has in-degree and out-degree Г. Then mutating at vertex results in quiver of Type I-Il (cf. Figure 12).
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
V1
U2
V3
V2
Re
U3 U4
U1
U4
Figure 12, Performing arr on a quiver of Type VT' (left).
Remark D.8. Our characterization was achieved independently, without knowledge of that of Henrich (2011), I). Although we will not discuss the overlap in detail, we give a brief description of how our characterization corresponds to (Henrich, 2011), using the notation found therein:
D...' in Henrich (2011) corresponds to our paired types, except when (+,*') = ((O,n). (O, m)) and Q is in so-called "merged stage". We call this type VI.
D(O,)aa D(O.))ee D P(OM) our types Va, V, Vb, respectively.
DEAD and Dena are our types V and Vb'. Our type Va' falls into the definition of D(o,a)a--
Dg is our type VT'
E. Proof of Theorem 6.1
Proof. From the preceding lemmas we know that these types are mutation equivalent the quiver in (7), so need only prove that these types are exhaustive by showing that M is closed under quiver mutation.
We will begin with the paired types. Suppose that Q E M is the union of two quivers Q1, E M whose intersection Te is in MA, If re E M for k > 1, then any mutation can affect the type of at most one of Q1 or Q2 and hence by Theorem 5.3 results a quiver of (possibly different) paired type. Thus in what follows we assume that Ie is a single vertex c. Moreover, we need only consider mutating at c, since a mutation anywhere else can only convert Q from one paired type to another.
In the casework below, when type V quiver has central cycle of length 3 and only 7 edges, will refer to it as minimal type V. Similarly, minimal type VI quiver is type VI quiver where both central cycles are length 3.
Type I-I. Because we assume F, is single vertex c, the underlying graph of this quiver is the star graph on 5 vertices, rooted at Mutating at с depends on the number of arrows to and from c. If has indegree 4 or outdegree 4, then Me simply reverses every arrow. e has indegree 3 or outdegree 3, then He produces a minimal Type V quiver. If c has indegree 2 and outdegree 2, then 44 produces Type VI (or VT', since they are the same when there are only vertices).
Type I-II. Let ç' denote the connecting vertex opposite e in the block Biv in the Type subquiver, and a, b denote the endpoints of the Type I arrows. Then if Q contains the paths then Me yields another Type I-II quiver. If Q has + d' with - and + b, or if Q has é' + c with a and b - the result is Type VI'. If Q has + e and c >> b (or vice versa) then the result is Type V (regardless of the orientation of the Biv block.
- c' and b - c-> e' or d' +
and d' - c b,
Type I-III. If the Type arrows are of the same orientation with respect to c, then Me results in a quiver of Type V. Otherwise, the result is Type VI.
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Type I-IV. If the Type arrows are of the same orientation with respect to c, then Me results in a quiver of Type V. Otherwise, the result is Type VI.
Type IIIII. Let c', c" denote the connecting vertices opposite e in the Biv blocks in the Type II subquivers Qi and Q2. respectively. Then if the arrows are oriented c' + + c" (or the reverse) then Me yields a quiver of Type VT where the connecting vertex è is glues the By block to an oriented triangle. Otherwise Ae yields another Type II-II quiver.
Type I-III. Mutating at e yields a Type V quiver (regardless of the с c- orientation).
Type II-I. Mutating at e yields Type V quiver (regardless of the c c- c' orientation).
Type IIIIIII. Mutating at yields Type VI quiver with two central cycles of length 3.
Type III-IV. Mutating at e yields Type VI quiver.
Type IV-IV. Mutating at c yields a Type VI quiver.
Having finished the paired types, we turn our attention to our newly identified types.
Type V. In the proof of Lemma D.I, we showed that mutating at a stays in 1- a quiver of paired type in all cases. In Appendix D.I, we showed that mutating at (and d' by symmetry) produces a Type Va quiver. If we mutate at b, we bifureate into the same cases as when mutating at a in Lemma D.I, and in fact obtain the same types. Now, the central cycle is of length 3, then we are done, as mutating anywhere along the central cycle will simply shrink the central cycle by 1, leaving with another quiver of Type V. However, suppose the central cycle is of length 3, given by S b + - Then we must consider (e,, which yields Type V'. (In this manner, Type V can be seen as the result of shrinking the central cycle to length 2, which we then remove because digons are prohibited.) Here the subquiver I' is a single vertex if there are no additional spikes, a single directed edge if there is one spike, and an oriented triangle if there are two spikes.
Type Va. We know from Appendix D.1 that (4a and Ma yield Types V and Vb, respectively. Continuing, H4a and (4 both yield Type VI. If the central cycle (sans a) is length > 3, then we are done. Otherwise, we consider the case where we have a vertex e with b > a and compute Pe, which we yields Type Va'.
Type Vb. From Appendix D.I we see that μa and pla yield Type Va. Mutating or b yields Type Vb again, simply moving the reversed arrow around the central cycle with the associated d, d'. Finally, we are done unless the central cycle is an unoriented triangle +- b - u -> a, in which case we must consider ftv, which yields Type Vb'.
Type VA. We know the mutations paa and pa yield Type Va'. The mutations pa and μb yield Type VT' Finally, we know that mutating e yields Type from Corollary D.3 (see Figure 11).
Type Va'. As we have seen, jaa yields Type V and pa yields Type Vb'. The mutations p4a and μs both result in Type Va' by eyclically permuting (a, b, d) forwards and backwards, respectively, Finally, Me yields Type Va.
Type Vb'. The mutations Ha and μa' yield Type Va'. Mutating at a or b produces an automorphism which swaps and d with and d', respectively, so Pa and 4b yield Type Vb' again. Finally, mutating at c yields Type Vb.
Type VI. From Lemma D.5 we know that jr. yields a paired type. Call the two central cycles C1 and C2. If we mutate at any vertex which is not adjacent to e, the result is still Type VI, as the mutation affects the relevant cycle C1 or C2 as a Type IV subquiver, and cannot break C1 or C2. Suppose then that we mutate at vertex v which is adjacent to and suppose, without loss of generality, that v E C1. Then pe simply moves e from C1 to C2, resulting in Type VI, unless C1 is a triangle, in which case the result is Type Va.
Type VT. Since e is connecting vertex in I, it has degree at most 2. If e has degree in F, Me yields Type I-I1, and if c has degree in I, Me yields Type II-II. Any other mutation in I' cannot change the type, so it remains only to check the vertices adjacent to c. Mutating at any results in Type V*.
Thus we have shown that MD_1 is closed under quiver mutation. This completes the proof.
F. A Aditioona figures
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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
Type I-I.
Type I-III.
r
r
Type I-II.
Type I-IV.
r
F
Q'
Type II-II.
F
Type IIIIII.
r
Type II-IV.
r
Q'
Type IIIIIII
r
M
Type III-IV.
F
ạ
Type IV-IV.
Q"
Figure 13, All paired types. Unoriented edges may have orientation. Cireles indicate oriented cycles. Here I. 1" are subquivers Type A, and Q' a subquiver Type D-IV for which c', or c" part spike. Notice that we may I E MMf with d. but Q' and Q" must contain at least two edges in addition to the ones shown.
21

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Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes
64
54
02
Duver
G8ver137534
Oueer 335325,
Ouve 13511,
Figure 14. Randomly selected quivers from the orange (left) cluster in Figure 8, consisting of quivers of Types V, Va, Vb, V Va', Vb'.
14
54
82
54
32
40
Figure 15, Randomly selected quivers from the blue (right) cluster in Figure 8, which consists of quivers of Types VI and VI'.
22


Reviews:
[{"review_id": "gKzNXEM3TC", "reviewer": "Reviewer_ivH2", "summary": "The paper uses GNNs to learn quiver mutation equivalence. The results show that GNNs can not only classify these quiver types, but can also characterize particular mutation classes through their latent representations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review, and we are glad you found the application to quiver mutation interesting. We hope to address your comments below:\n\n> As admitted in the paper, the gap between train and test set (including the distribution of the number of nodes, and the absence of $\\widetilde{E}$ in the test set) does not follow common machine learning settings. All the experiments are carried on one dataset, while more comprehensive will be more convincing.\n\nRegarding the experimental methods, we recognize that the difference in the train and test sets does not follow common machine learning practice. Rather, *our experimental design is motivated by our ultimate goal of recovering classification theorems from the model*. As we discuss in our Response to Reviewer KQ27, the presence of $\\widetilde{E}$ in the training set serves to modulate the difficulty of the problem to ensure that the model learns discriminative features for the other classes, particularly $D$ and $\\widetilde{D}$, on which we focus our explainability study. Similarly, the difference in the distribution in the number of nodes between training and testing ensures that the GNN learns structural traits which are size-generalizable, as well as allowing us to provide the GNN with the full mutation classes for smaller sizes without contaminating the test data.\n\n> The paper only uses one type of classical GNN architecture, without investigating more expressive and more state-of-the-art GNN variants. Simultaneously, the explanation method only involves PGExplainer, while there are also many other GNN explainers.\n\nOur intention was not to investigate which GNN architecture would be the most effective for classifying quiver mutation classes, nor was it to compare between different explainability methods. Rather, *we provide a case study that illustrates how a domain expert using off-the-shelf components can guide their own research with machine learning*. This application-driven approach differs from works whose goal is to introduce novel architectures or algorithms to be adopted by others. In such settings we agree that it is important to show that the architecture will perform well beyond the initial dataset it was trained for. In our work, however, once the model offers sufficient insight to conjecture and prove a statement about the quiver mutation classes of interest, its performance on additional data ceases to be important.\n\nUltimately, while we admit the individual methods are not novel in and of themselves, we believe our work presents a novel application of existing methods to an interesting problem in combinatorics. More broadly, we believe the central idea that machine learning can guide domain research is absolutely of interest to the ICML audience.\n\n> Can you provide analysis on theoretical expressive power of directed GNNs, or at least for distinguishing quiver mutation classes?\n\nThe question of GNN expressive power (equivalently, the WL test) and quiver mutation is interesting, and we have expanded this discussion in the paper. Directed GNNs are known to be strictly more expressive than undirected 1-WL [1], and we thank you for pointing out this oversight in our discussion and references. In the context of quiver mutation, there are some relevant local substructures which an undirected GNN clearly cannot distinguish. For example, a directed triangle should be of Type $A_3$ while an undirected triangle is of Type $\\tilde{A}_2$. In general, the directed WL test is sufficient to recognize type $A$ quivers, as well as certain subtypes of $D$ and $\\widetilde{D}$ quivers (namely Types I, II and the corresponding paired types). However, one of the WL test's limitations is its inability to count larger cycles, such as those that appear in Type $D$-IV and $\\widetilde{D}$-VI.\n\n> Can you elaborate on more large-scale/real-world datasets and more state-of-the-art models?\n\nGNNs' ability to generalize across sizes and their inductive bias towards local structures also motivates our choice of architecture. While e.g. graph transformers also provide promising reasoning capabilities on graph tasks, message-passing remains the state of the art for recognizing many of the local substructures which are relevant to this problem [2]. However, as we note above, message-passing fails to recognize larger-scale structures in graphs. Our response to Reviewer E1Ro contains an empirical investigation on larger graphs (up to $n = 20$ nodes), revealing that our GNN generalizes well but not perfectly on larger graphs.\n\n[1] Beddar-Wiesing et al., On the Extension of the Weisfeiler-Lehman Hierarchy by WL Tests for Arbitrary Graphs, MLG@ECMLPKDD 2022\n\n[2] Sanford et al., Understanding Transformer Reasoning Capabilities via Graph Algorithms, NeurIPS 2024", "claims_and_evidence": "Most claims are supported by evidence including theoretical results, experiments or case analysis.", "methods_and_evaluation_criteria": "As admitted in the paper, the gap between train and test set (including the distribution of the number of nodes, and the absence of $\\tilde E$ in the test set) does not follow common machine learning settings. All the experiments are carried on one dataset, while more comprehensive will be more convincing.", "theoretical_claims": "I check the correctness and did not find major flaws. However, a great proportion of definitions (those in Section 3) and theoretical results (e.g., Theorem 4.3) are known results or trivial extensions, which bothers reading and may downgrade the contribution of the paper. While there are tons of theoretical work on the expressivity of graph neural networks, I strongly recommend the authors look deeper into theoretical expressive power of (directed) GNNs to provide theoretical guarantees for GNNs to characterize quiver mutation classes.", "experimental_designs_or_analyses": "The paper only uses one type of classical GNN architecture, without investigating more expressive and more state-of-the-art GNN variants. Simultaneously, the explanation method only involves PGExplainer, while there are also many other GNN explainers. Moreover, the small model size and inadequate training does not match usual deep learning schemes, which make the results less convincing.", "relation_to_broader_scientific_literature": "The paper is related to literature in mathematics, topology and graph neural networks.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is an interesting application of GNNs to the mutation equivalence problem. However, the machine learning techniques in this paper are elementary, and the analytical results lack statistical guarantee over larger scale.", "other_comments_or_suggestions": "This paper is essentially a naïve application of simple machine learning methods to one specific problem, without strong theoretical guarantees and large scale experimental verifications. My opinion is that it is not yet qualified for a top-tier AI/machine learning conference like ICML, and would suggest submitting the paper to some other conferences/journals for data analysis or applied mathematics.", "questions_for_authors": "* Can you provide analysis on theoretical expressive power of directed GNNs, or at least for distinguishing quiver mutation classes?\n\n* Can you elaborate on more large-scale/real-world  datasets and more state-of-the-art models?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "27KJAsOM8o", "reviewer": "Reviewer_KQ27", "summary": "This paper shows that a GNN learns the same substructure to classify the quiver mutation class of a quiver as proposed by a classification theorem in quiver theory in mathematics. The authors train a GNN on quivers of different types $A,D,E,\\tilde{A},\\tilde{D},\\tilde{E}$ and use PGExplainer on this trained model to detect the substructure that the model is focussing on, in order to classify the mutation class of quivers. Further, in section 5 they conjecture a theorem which they prove about the mutation class of quivers of type $\\tilde{D}$.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your insightful review. We are encouraged you think our work demonstrates the promise of machine-guided research, and we are glad you found the paper well-organized. Since multiple reviewers had questions about the experimental design, we have also expanded the discussion in the paper to address the most common questions and emphasize that our experimental design was driven by our ultimate goal of generating domain-related insight from the model. \n\nWe aim to address your questions more specifically below:\n\n> Why do you choose to train on A,D,E, $\\widetilde{A}$, $\\widetilde{D}$, $\\widetilde{E}$, when the test set is not going to contain any samples from $\\widetilde{E}$? \n\nWe train the model on entire mutation classes of smaller sizes to ensure the model sees a comprehensive view of each mutation class during training. Our inclusion of types $E$ and $\\widetilde{E}$ in the train set primarily serves to increase the difficulty of the training task, since ultimately our analysis focuses on types $D$ and $\\widetilde{D}$. (From the domain perspective, there are only finitely many $\\widetilde{E}$ of any size, so general classification can be achieved through exhaustive computation, and the class $E$ is no longer mutation-finite for larger sizes, making a combinatorial characterization substantially more difficult.)\n\n> Moreover, what is the reason behind training on quivers with lesser nodes and testing on quivers of a fixed node size which is larger than what the model has been trained on?\n\nThe difference in sizes from train to test is in part a consequence of our desire to train on entire mutation equivalence classes, as we mentioned previously. In addition, from the application perspective any classification result for a mutation class should generalize across sizes. In this manner the difference in graph sizes follows prior work studying size generalization of GNNs (e.g. [1]).\n\n> You might want to add the fact that the message passing is being done over graphs with constant node features to highlight the fact that the GNN model is truly learning only the structure with little to no help from the node features.\n\nThank you, this is an insightful suggestion! We have added this to the paper.\n\n> Did you try generating the graphs with different features? E.g. Node degrees or similar features?\n\nWe did not generate graphs with node features. As you point out, we used constant node features so that the GNN learned entirely from the graph structure. In particular, our ultimate goal was to provide structural characterizations of mutation classes, so we wished to ensure that the GNN was truly relying on the graph structure to perform classification.\n\n[1] Yehudai et al., From Local Structures to Size Generalization in Graph Neural Networks, ICML 2021", "claims_and_evidence": "The claims made in the paper are supported by sufficient evidence.", "methods_and_evaluation_criteria": "A holistic viewpoint of the methods seems to make sense for the problem at hand. However, some finer details such as choice of design of experiments do not seem to be motivated enough, as pointed out in the later section of the review.", "theoretical_claims": "Yes, I checked the proof of Theorem 5.1, and as a consequence Lemmas D.1, D.5-D.7 and Corollaries D.2 - D.4 in Appendix D.", "experimental_designs_or_analyses": "Why do you choose to train on A,D,E,$\\tilde{A}$,$\\tilde{D}$,$\\tilde{E}$ when the test set is not going to contain any samples from $\\tilde{E}$?\n\nMoreover, what is the reason behind training on quivers with lesser nodes and testing on quivers of a fixed node size which is larger than what the model has been trained on?", "relation_to_broader_scientific_literature": "I think this work shows that machine-guided research is an avenue that needs to be explored more by the AI for science community. Though the specific task that the authors choose to show this might seem niche, the idea that machine recognizes the same substructures in a problem that a classification theorem gives is really powerful.", "essential_references_not_discussed": "I don't think so.", "other_strengths_and_weaknesses": "Strengths:\n\nPaper is well-written and well-organized.\n\nWeaknesses:\n\nAs pointed out earlier, the choice of the design of experiments does not seem to be motivated enough.", "other_comments_or_suggestions": "You might want to add the fact that the message passing is being done over graphs with constant node features to highlight the fact that the GNN model is truly learning only the structure with little to no help from the node features.", "questions_for_authors": "Did you try generating the graphs with different features? E.g. Node degrees or similar features?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6r9JS680rf", "reviewer": "Reviewer_WJi6", "summary": "This work uses GNNs to solve the quiver-mutation-equivalence problem: whether one quiver can be transformed into the other through a sequence of mutations. With explainability techniques, they discovers criteria for quiver of type $\\tilde D$. Moreover, GNN need not to be trained.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your feedback and your suggestions regarding presentation. Below we hope to address your concerns and questions.\n\n> This work is more similar to a experiment report rather than a paper, as no novel algorithm or tasks are proposed, and the result is used to verified existing results rather than new results.\n\nAlthough our work is an application-driven case study rather than a specific novel method, we believe the application of existing machine learning techniques to the quiver mutation equivalence problem is novel and has potential impact in AI for mathematics. While our mutation equivalence results were known, we reiterate that *we discovered Theorem 5.1 independently of prior work*.\n\nAs we have also discussed in or response to Reviewer ivH2, the use of ML to accelarate scientific discovery is of great interest to the machine learning community. Indeed, it is often pointed to as one of the primary positive impacts that modern machine learning can provide. Understanding how ML tools should be used to enable scientific discovery must be a conversation that involves both domain and ML experts. As such, we believe that works that illuminate this process are very much in line with the role of ICML in the ML community.\n\n> In line 157, please refer to figure 3 for a more clear illustration of quiver type A,D,E ...\n\nThank you for the suggestion. We will move the reference to Figure 3 from line 142 to line 157.\n\n> Equivalence relation is more similar to a contrastive learning task rather than classification task. Have you even tried formulate the task as predicting whether two graph are equivalent rather than predict the type of one graph?\n\nThe contrastive learning task is an interesting question in its own right. However, our goal of extracting structural characterization results from the model motivated our choice to formulate the task as classification. From an explainability perspective, we are interested in the question \"Why does the GNN classify this as Type $D$?\" rather than the question \"Why does the GNN predict these two quivers are mutation equivalent?\" Thus we chose to formulate the machine learning task as a classification problem rather than a contrastive problem.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes. The proposed GNN is a simple adaptation of GIN to directed multigraph settings.", "theoretical_claims": "No. All proofs are on the property of quiver mutations, which are primary the background problem setting rather than the method. I do not have the domain knowledge and skip it.", "experimental_designs_or_analyses": "Yes.  The training set contains quiver with nodes 6-10, while test set containing quiver with 11 nodes only, which avoids the risk of data leakage.", "relation_to_broader_scientific_literature": "As the author claims, it is related to the quiver mutation problem.", "essential_references_not_discussed": "The GNN and Explainablity methods used in the paper are discussed in method or Appendix section, but not introduced in related work section.", "other_strengths_and_weaknesses": "This work is more similar to a experiment report rather than a paper, as no novel algorithm or tasks are proposed, and the result is used to verified existing results rather than new results.", "other_comments_or_suggestions": "In line 157, please refer to figure 3 for a more clear illustration of quiver type A,D,E ...", "questions_for_authors": "Equivalence relation is more similar to a contrastive learning task rather than classification task. Have you even tried formulate the task as predicting whether two graph are equivalent rather than predict the type of one graph?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eSWlXI8gpm", "reviewer": "Reviewer_E1Ro", "summary": "In this paper, the authors study the problem of quiver mutation using the GNNs and GNN explanation tool. The author identified that the GNNs trained with naive classification tasks on predicting quiver mutation type are able to learn causal information related to quiver mutation. In particular, it can identify some existing theorems without any prior knowledge.\n\n## Update after rebuttal\nI would like to thank the authors for the detailed response to my questions and concerns. Most of my concerns are appropriately addressed by the rebuttal. I would like to keep my original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback and insightful questions. We're glad to hear you found the topic interesting, and we hope to answer your questions below:\n\n> Is it possible to evaluate on a graph with a size much larger than training (like 20 or 30), and will the conclusions still hold? \n\nThe properties of each mutation equivalence class remain valid *regardless of the size*, so from the domain perspective the exact size of the graphs in the test set is not particularly important.\n\nFrom the machine learning perspective, we can certainly evaluate the model on larger graphs. Below we test the same model checkpoint on quivers of up to 20 nodes. Because the number of distinct quivers grows very quickly with size, we can only check on a subsample (we use a mutation depth of 6 and, if necessary, randomly sample 100,000 quivers to avoid out-of-memory errors).\n\n| Nodes | Accuracy |\n|:-----:|:--------:|\n| 12    | 99.6%    |\n| 13    | 98.7%    |\n| 14    | 97.7%    |\n| 15    | 95.5%    |\n| 16    | 94.3%    |\n| 17    | 92.0%    |\n| 18    | 91.1%    |\n| 19    | 89.4%    |\n| 20    | 89.1%    |\n\nThe model continues to perform well though performance does begin to degrade. We believe this is because the GNN has a fixed depth of 4, and hence may struggle to capture certain larger non-local substructures that appear in quivers with more nodes (e.g. long cycles). We discuss the theoretical implications of using a message-passing GNN for large quivers in the Response to Reviewer ivH2.\n\n> The analysis only focuses on the type $D$ and $\\widetilde{D}$, Is it possible to also extend the analysis to other types and discover other patterns?\n\nIt should be possible to extract the characterizations of other finite mutation classes in a similar manner, recovering other results from Henrich [1]. Certain mutation-infinite classes may also admit characterization by certain patterns, though it is not known in general which other mutation classes, if any, admit such a characterization. If such a characterization does exist, it may be possible to extract these from a machine learning model as well.\nHowever, we note that mutation equivalence is in general a very hard problem [2], so discovering patterns for an arbitrary mutation class is likely also difficult in general.\n\n> I think how to systematically classify a certain problem to corresponding ML class that is algorithmic alienable, how to train the ML model and how to leverage explanation tool to discover new knowledge would be more interesting and impactful.\n\nWe agree, and we think this is an interesting direction for future work. GNNs are a natural choice for many problems, but there is emerging evidence for reasoning capabilities in transformers, for example. A more general systematic pipeline, however, will require a more mature suite of interpretability tools and a deeper understanding of algorithmic alignment in different architectures.\n\n[1] Henrich, Mutation-classes of diagrams via infinite graphs, Math Nachr., 2011\n[2] Soukup, Complexity of quiver mutation equivalence, preprint, 2023", "claims_and_evidence": "The overall procedure makes sense to me, and using the algorithm alignment ability of GNNs to solve mathematic problems is interesting.", "methods_and_evaluation_criteria": "1. I am not really familiar with the quiver mutation. But I am just wondering if there is any additional concern that the dataset only contains graphs of size 6-10 in training and 11 for testing. Is it possible to evaluate on a graph with a size much larger than training (like 20 or 30), and will the conclusions still hold? \n2. The analysis only focuses on the type $D$ and $\\widetilde{D}$, Is it possible to also extend the analysis to other types and discover other patterns?\n3. The paper only examines several particular theorems. I am wondering how we can generalize results to other problems, especially for problems where we do not know whether GNN is useful. As the finally goal is that we can use ML models to solve mathematical problem or identify theorem we are unknown, I think how to systematically classify a certain problem to corresponding ML class that is algorithmic alienable, how to train the ML model and how to leverage explanation tool to discover new knowledge would be more interesting and impactful.", "theoretical_claims": "As I am not familiar with Quiver mutation, I cannot assess the most theoretical claims made in the paper.", "experimental_designs_or_analyses": "See above.", "relation_to_broader_scientific_literature": "The method and pipeline used in the paper may be applicable to other mathematic problems.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "See above", "other_comments_or_suggestions": "See above", "questions_for_authors": "See above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper uses GNNs to learn quiver mutation equivalence. The results show that GNNs can not only classify these quiver types, but can also characterize particular mutation classes through their latent representations.

The reviewers agree that the paper meets the standards of an Application-Driven Machine Learning Paper: novel methods, datasets, tasks, and/or metrics. The reviewers state that the paper is  a well-motivated application paper for applying GNNs to a math problem, and that the authors provide many experimental results to demonstrate that well-trained GNNs can identify existing mathematical results automatically. Moreover, during the rebuttal, the authors also provided additional experiments on the generalization results, which are positive support for their existing results. The authors are asked to include these results in their camera-ready paper if accepted. 

The reviewers also state two main limiting factors in the paper: 1) the paper only provides evidence for using GNNs to validate existing results, but does not discover any new scientific results, and 2) the potential audience is limited by the fact that the problem under investigation is somewhat esoteric. Still the very fact that GNNs can discover mathematical results is worthy of publication.
Decision: Accept (poster)