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

TL;DR

The paper addresses expressivity limits of neighborhood-based GNNs by introducing covers as algebraic graph representations. It develops the Grothendieck Graph Neural Networks (GGNN) framework, which builds covers from directed subgraphs into a monoid $\mathsf{Mod}(G)$, maps them to matrices via a monoidal homomorphism $\mathsf{Tr}$, and enables topology-aware message passing. As an instantiation, the Sieve Neural Network (SNN) leverages a cover of sieves to capture richer topological patterns, achieving strong performance on graph isomorphism benchmarks and TUDataset tasks while preserving invariance under isomorphism. The framework offers a flexible, mathematically grounded path to design diverse GNNs that go beyond WL expressivity and can be integrated with existing GNN paradigms, with potential impact on graph representation learning and structural graph analysis.

Abstract

Due to the structural limitations of Graph Neural Networks (GNNs), in particular with respect to conventional neighborhoods, alternative aggregation strategies have recently been investigated. This paper investigates graph structure in message passing, aimed to incorporate topological characteristics. While the simplicity of neighborhoods remains alluring, we propose a novel perspective by introducing the concept of 'cover' as a generalization of neighborhoods. We design the Grothendieck Graph Neural Networks (GGNN) framework, offering an algebraic platform for creating and refining diverse covers for graphs. This framework translates covers into matrix forms, such as the adjacency matrix, expanding the scope of designing GNN models based on desired message-passing strategies. Leveraging algebraic tools, GGNN facilitates the creation of models that outperform traditional approaches. Based on the GGNN framework, we propose Sieve Neural Networks (SNN), a new GNN model that leverages the notion of sieves from category theory. SNN demonstrates outstanding performance in experiments, particularly on benchmarks designed to test the expressivity of GNNs, and exemplifies the versatility of GGNN in generating novel architectures.

Figures (4)

• Figure 1: Here are two examples of directed subgraphs, $\hat{D}$ in the middle and $\bar{D}$ on the right, of a graph $G$ on the left. The directed subgraphs $\hat{D}$ and $\bar{D}$ can be considered as strategies to broadcast the messages in the graph $G$.
• Figure 2: Visualizing a neighborhood by representing it as a directed subgraph
• Figure 3: Left: A graph $G$. Right: $\mathsf{Sieve}(v,3)=D_3(v)\bullet D_2(v)\bullet D_1(v)$. Directed edges in yellow, red and black specify $D_1(v)$, $D_2(v)$ and $D_3(v)$ recpectively
• Figure 5: The graph $G$, the left one, and $H$, the right one, are not distinguishable by MPNN

Theorems & Definitions (43)

• Definition 3.0.1
• Definition 3.0.2
• Proposition 3.0.1
• Example 3.0.1
• Theorem 3.1
• Theorem 3.2
• Example 3.2.1
• Definition 3.2.1
• Definition 3.2.2
• Theorem 3.3