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

TL;DR

The paper introduces GkGNN, an algebraic framework that replaces neighborhood primitives in GNNs with covers, enabling topology-aware message passing and improving expressivity on graph isomorphism tasks.

Contribution

It proposes a novel algebraic extension using covers, formalizes them into matrices, and designs Sieve Neural Networks that outperform traditional neighborhood-based GNNs.

Findings

SNN achieves zero failures on challenging graph isomorphism benchmarks.

GkGNN provides a principled foundation for topology-aware GNNs.

Experiments demonstrate improved evaluation via label propagation.

Abstract

Graph Neural Networks (GNNs) are almost universally built on a single primitive: the neighborhood. Regardless of architectural variations, message passing ultimately aggregates over neighborhoods, which intrinsically limits expressivity and often yields power no stronger than the Weisfeiler-Lehman (WL) test. In this work, we challenge this primitive. We introduce the Grothendieck Graph Neural Networks (GkGNN) framework, which provides a strict algebraic extension of neighborhoods to covers, and in doing so replaces neighborhoods as the fundamental objects of message passing. Neighborhoods and adjacency matrices are recovered as special cases, while covers enable a principled and flexible foundation for defining topology-aware propagation schemes. GkGNN formalizes covers and systematically translates them into matrices, analogously to how adjacency matrices encode neighborhoods, enabling both theoretical analysis and practical implementation. Within this framework, we introduce the cover of sieves, inspired by category theory, which captures rich topological structure. Based on this cover, we design Sieve Neural Networks (SNN), a canonical fixed-cover instantiation that generalizes the adjacency matrix. Experiments show that SNN achieves zero observed failures on challenging graph isomorphism benchmarks (SRG, CSL, BREC) and substantially improves topology-aware evaluation via a controlled label-propagation probe. These results establish GkGNN as a principled foundational framework for replacing neighborhoods in GNNs.