Grothendieck Graph Neural Networks Framework: An Algebraic Platform for Crafting Topology-Aware GNNs

Reply to Reviewer REnp

Thank you for your valuable feedback. Below, we address your comments and questions in detail. $\textbf{Responses to Weaknesses}$ $\textbf{1. Comparisons with Related Methods}$ We appreciate your suggestion to compare GGNN with other methods, such as path-based GNNs, substructure-based GNNs, and $k$-hop message-passing GNNs. These methods also integrate higher-order structural information, but GGNN distinguishes itself by providing a fundamentally new perspective through its algebraic framework. Specifically: $\textbf{Novel Representations for Graphs:}$ While other methods rely on the adjacency matrix as the foundational representation of graphs, GGNN provides infinitely many alternatives to the adjacency matrix, thereby expanding the design space for GNN models. $\textbf{Algebraic Tools:}$ The framework introduces algebraic structures such as the monoid $\mathsf{Mod}(G)$ and the binary operation $\bullet$, which allows defining alternative concept for neighborhoods called covers. These covers serve as flexible alternatives to traditional neighborhoods. $\textbf{Expressive Power:}$ Theorems 2.4.1 and 2.4.2 in the paper prove that GGNN offers a $\textit{unique up-to-isomorphism algebraic description}$ for graphs. As a result, GGNN can always distinguish between non-isomorphic graphs by choosing appropriate covers, offering a theoretical guarantee not provided by many existing methods. $\textbf{2. Readability Improvements}$ We acknowledge the need for improved clarity in presenting the framework. To enhance readability, we will: Add a $\textbf{figurative example}$ illustrating how $\mathsf{SNN}$ distinguishes strongly regular graphs, specifically those that the 3-WL test fails to differentiate. Incorporate background information on $\textbf{category theory}$ in the preliminaries to make the paper more self-contained and accessible to readers unfamiliar with this topic. Present the results on the SR and CSL datasets in a clear $\textbf{tabular format}$. $\textbf{Response to Questions}$ $\textbf{Advantages of GGNN:}$ GGNN introduces a fundamentally new way of viewing graph neighborhoods by leveraging algebraic structures: The monoid $\mathsf{Mod}(G)$ provides infinitely many alternatives to traditional neighborhoods, allowing researchers to consider graphs from diverse perspectives tailored to specific goals. GGNN introduces the monoidal homomorphism $\mathsf{Tr}$, which transforms covers into collections of matrices as alternatives to the adjacency matrix. These diverse numerical representations can be used as input for other GNN models. Theorems 2.4.1 and 2.4.2 prove that GGNN gives a $\textit{unique up-to-isomorphism description}$ of a graph, enabling it to distinguish any two non-isomorphic graphs. $\textbf{Advantages of}$ $\mathsf{SNN}$: $\mathsf{SNN}$ exemplifies how GGNN can be applied to design a topology-aware GNN model: $\mathsf{SNN}$ is specifically designed to extract deep topological features of graphs. For example, in Example B.0.2, it distinguishes between holes formed by 3 edges and those formed by 4 edges. Experiments demonstrate $\mathsf{SNN}$'s ability to differentiate strongly regular graphs, achieving $100$ percent accuracy on these datasets. This highlights its capacity to handle one of the most challenging graph isomorphism tasks. $\mathsf{SNN}$ resists oversmoothing by considering all paths of specific lengths between nodes during message passing. As a result, even distant nodes can exchange information effectively. We hope this response addresses your concerns and provides a clearer understanding of the contributions and strengths of GGNN and $\mathsf{SNN}$.

Reply to Reviewer dtWY

Thank you for taking the time to review our paper. $\textbf{Main Contribution Clarification}$ The GGNN framework provides a systematic improvement in the GNN literature by addressing a fundamental question: *Where does the concept of neighborhood originate, and does this origin allow for alternative definitions?* To explore this question, we developed an algebraic theory that naturally involves new terms and definitions. Below, we clarify the framework’s contributions. In this paper, the origin of the neighborhood concept is formalized as a monoid $\mathsf{Mod}(G)$, which contains infinitely many elements. The alternatives to neighborhoods are presented as collections of finitely many elements of this monoid, referred to as *covers*. The GGNN framework provides algebraic tools, particularly the binary operation $\bullet$, to construct diverse covers. We further proved that all elements of $\mathsf{Mod}(G)$ can be generated using directed edges and the operation $\bullet$. Additionally, the framework introduces a monoidal homomorphism $\mathsf{Tr}$, which transforms covers into collections of matrices, serving as alternatives to adjacency matrices. To demonstrate the practical utility of this framework, we introduced the $\mathsf{SNN}$ model. This example illustrates how to leverage the tools in GGNN to design a GNN model. The construction of $\mathsf{SNN}$ showcases the process of utilizing the framework to create models tailored to specific objectives. $\textbf{Responses to Specific Comments}$ $\textbf{1. Directed Subgraphs and Their Role}$ The definition of directed subgraphs as a basis for the monoid $\mathsf{Mod}(G)$ is carefully chosen and plays a fundamental role in the theory. While we understand the suggestion to incorporate acyclic graphs, this substitution would disrupt the theoretical foundation and is therefore not viable. $\textbf{2. Clarification on $\mathsf{Rep}(D)$}$ As illustrated in Figure 1, $\mathsf{Rep}(D)$ is not equivalent to a directed adjacency matrix. Instead, it encodes critical path information within a directed subgraph, making it more expressive than a standard adjacency matrix. We will elaborate on this distinction in the revised version. $\textbf{3. Experimental Results on SR and CSL}$ In the experimental section, we stated that $\mathsf{SNN}$ achieves perfect differentiation between all graphs in the strongly regular (SR) and CSL collections, resulting in an accuracy of $100$ percent. We appreciate your suggestion and will include a detailed table to present these results more clearly in the revised manuscript. $\textbf{4. Related Work}$ We acknowledge the limited discussion of related work and will update this section to include additional references, such as [1]. $\textbf{Answers to Reviewer Questions}$ $\textbf{Q1: Theoretical Superiority and Complexity Trade-off}$ GGNN is a framework rather than a standalone model. Its primary advantage lies in its ability to provide algebraic tools for designing GNN models tailored to specific tasks. For example, $\mathsf{SNN}$ demonstrates how GGNN enables the extraction of rich topological information. Its outstanding performance on strongly regular graph collections highlights this capability. Furthermore, $\mathsf{SNN}$ exemplifies how concepts from other fields (e.g., sieves from category theory) can be integrated into GNN model design via GGNN. Each model created using GGNN has its own complexity, depending on its specific design and purpose. The complexity of a specific instance (e.g., $\mathsf{SNN}$) should not be generalized as a limitation of the framework itself. For $\mathsf{SNN}$, the time complexity for sparse graphs is $\mathcal{O}(|E| \cdot |V|^2)$. This complexity is justified by the model’s focus on extracting detailed topological information. $\textbf{Q2: Directed Subgraph Generation and GGNN Power}$ According to Theorem 2.2.2, appropriate monoidal elements can be generated using directed edges and the binary operation $\bullet$. This flexibility allows researchers to define directed subgraphs or monoidal elements according to their objectives. The GGNN framework does not impose constraints on these choices, enabling it to support a wide range of use cases. For examples of models and their corresponding directed subgraphs, please refer to the examples provided in our response to Reviewer kNJA. It is also worth noting that $\mathsf{SNN}$ is used as a preprocessing tool to transform datasets by replacing adjacency matrices with its outputs. Once the transformation is complete, a chosen message-passing operator can be applied, minimizing the scalability concerns during training. We hope these clarifications address your concerns and provide a better understanding of the GGNN framework and its contributions.

Reply to Reviewer wDej

Thank you for taking the time to provide valuable feedback on our work. We address your concerns below. $\textbf{Main Contribution Clarification}$ As stated in the paper, our primary contribution lies in algebraically extending the concept of neighborhoods in graphs. We achieve this through the introduction of the GGNN framework, which provides a foundation for creating novel GNN models. $\mathsf{SNN}$ serves as an example to illustrate the practical application of the GGNN framework. A significant portion of the paper (approximately 13 pages) is devoted to presenting a $\textbf{Strong Algebraic Theory}$ that explores and extends the concept of neighborhoods. The models, such as $\mathsf{SNN}$ and the examples provided in response to Reviewer kNJA, demonstrate the transformative capability of the GGNN framework. Each model has its own complexity, and the complexity of a specific instance (e.g., $\mathsf{SNN}$) should not be generalized as a limitation of the framework itself. $\textbf{Responses to Specific Comments}$ $\textbf{1. Scalability and Complexity}$ We emphasize that most real-world graphs are sparse. As noted in the paper, for sparse graphs, the time complexity of $\mathsf{SNN}$ reduces to $\mathcal{O}(|E| \cdot |V|^2)$. Since $\mathsf{SNN}$ is specifically designed to extract topological information from graphs, this complexity is justified by the richness of the information it captures. Furthermore, it is important to note that $\mathsf{SNN}$ is primarily used to preprocess datasets by replacing the adjacency matrix with its output. Once this transformation is complete, the training phase proceeds with any chosen message-passing operator, effectively mitigating scalability concerns during training. $\textbf{2. Experiments on Strongly Regular Graphs (SR) and CSL}$ In the experimental section, we demonstrated that $\mathsf{SNN}$ achieves perfect differentiation between all graphs in the strongly regular (SR) and CSL collections. This corresponds to an accuracy of $100$ percent. These results highlight the model’s ability to capture and utilize deep topological features of a graph. We will revise the text to make this point clearer and to include quantitative results for better clarity. $\textbf{3. Table 1 and Strong Baselines}$ We appreciate your observation regarding Table 1. The absence of certain strong baselines, such as [1], is noted, and we will revise the table to include comparisons with these methods to provide a more comprehensive evaluation. $\textbf{4. Missing Related Works}$ Thank you for pointing out the missing references, including [2], [3], [4], and [5]. We acknowledge the importance of discussing recent advancements and will update the related works section accordingly. This will ensure a broader and more current contextualization of our contributions. $\textbf{Minor Issue: Inappropriate Usage of Citations}$ We regret the inappropriate usage of \texttt{citet} and \texttt{citep} throughout the paper. This will be corrected in the revised version to improve the readability and professionalism of the manuscript. We hope that these responses address your concerns and help clarify our contributions.

Reply to Reviewer kNJA

Thank you for taking the time to provide insightful feedback on our work. Below, we address your concerns in detail. $\textbf{Main Contribution Clarification}$ The primary goal of this paper is to algebraically extend the notion of neighborhoods in graphs, enabling greater flexibility and efficiency while maintaining simplicity in practical applications. This extension is designed to seamlessly integrate covers into the construction of GNN models. The proposed GGNN framework serves as a versatile tool for developing new GNN architectures, demonstrated by the example of $\mathsf{SNN}$. $\textbf{Responses to Specific Comments}$ $\textbf{1. Examples of GNN Architectures Beyond}$ $\mathsf{SNN}$ To illustrate the utility of the GGNN framework in designing novel GNN architectures, we provide two additional examples: $\textbf{First Example:}$ We construct a cover different from the traditional neighborhood or the $\mathsf{SNN}$ cover by forming a collection of elements in the monoid $\mathsf{Mod}(G)$. Each node $v \in G$ is associated with the monoidal element $2\text{-}\mathrm{hop}(v)$: $$2\text{-}\mathrm{hop}(v):=({{\bullet}}_{ _{u\in N(v)}}S_u)\bullet S_v$$ Here, $S_u$ and $S_v$ represent the monoidal elements corresponding to $N(u)$ and $N(v)$, the neighborhood of $u$ and $v$ (see the proof of theorem 2.5.1). The transformation $\mathsf{Tr}$ maps this cover into a collection of matrices, providing an alternative for adjacency matrix. This approach can be generalized to $k\text{-}\mathrm{hop}(v)$. $\textbf{Second Example:}$ For a node $v \in G$, we select the highest-degree neighbors $v^1, \ldots, v^t \in N(v)$ and form the monoidal element $D_1^v$: $$D_1^v = e_1 \bullet \cdots \bullet e_t$$ where $e_i$ represents the directed edge from $v$ to $v^i$. In subsequent iteration, we exclude $v$ and recursively construct elements $D_1^{v^1}, \ldots, D_1^{v^t}$. After $k$ iterations, we aggregate the results to form $D_v$: $$D_v = D_1^v \bullet \cdots \bullet D_k^v$$ The transformation $\mathsf{Tr}$ converts these elements into matrix representations, offering an alternative for adjacency matrix that incorporates information about higher-degree nodes. These examples highlight how the GGNN framework translates diverse ideas into implementable GNN models, enabling flexible exploration of graph structures. $\textbf{2. Expressivity and Comparisons with}$ $k$-WL As demonstrated experimentally, $\mathsf{SNN}$ surpasses $3$-WL in distinguishing strongly regular graphs. It is empirically verified that $\mathsf{SNN}$ can resolve structural differences that $3$-WL cannot. We acknowledge the importance of theoretical analysis and are actively working on formalizing comparisons between $\mathsf{SNN}$ and $k$-WL in an ongoing study. $\textbf{3. Clarification of Main Contribution}$ While $\mathsf{SNN}$ exemplifies the GGNN framework in practice, our main contribution lies in the algebraic tools provided by GGNN. These tools facilitate the definition of covers, the use of monoidal structures, and the transformation $\mathsf{Tr}$, enabling systematic development of novel GNN architectures. This framework helps to get rid of conventional neighborhoods. $\textbf{4. Scalability and Performance}$ Most real-world graphs are sparse. As discussed in the paper, $\mathsf{SNN}$ has a time complexity of $\mathcal{O}(|E| \cdot |V|^2)$. In practical scenarios, as outlined in our experiments, $\mathsf{SNN}$ is primarily used to preprocess the dataset, transforming it into a new representation. Subsequently, any message-passing operator can be applied during the training phase, ensuring scalability and compatibility with existing frameworks. We sincerely hope these clarifications address your concerns and strengthen the understanding of our contributions. Thank you again for your valuable comments.