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

arXiv: https://arxiv.org/abs/2412.08835 · HTML: https://mlsanity.com/paper/2412.08835/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 31 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 24 of 24)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Combining Label Propagation and Simple Models Out-performs Graph Neural Networks](https://mlsanity.com/paper/2010.13993/) (2020)
- [Identity-aware Graph Neural Networks](https://mlsanity.com/paper/2101.10320/) (2021)
- [Graph Neural Tangent Kernel: Fusing Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/1905.13192/) (2019)
- [Relational Pooling for Graph Representations](https://mlsanity.com/paper/1903.02541/) (2019)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [A Survey on The Expressive Power of Graph Neural Networks](https://mlsanity.com/paper/2003.04078/) (2020)
- [Building powerful and equivariant graph neural networks with structural message-passing](https://mlsanity.com/paper/2006.15107/) (2020)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [A Comprehensive Survey on Graph Reduction: Sparsification, Coarsening, and Condensation](https://mlsanity.com/paper/2402.03358/) (2024)
- [KerGNNs: Interpretable Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/2201.00491/) (2022)
- [An Empirical Study of Realized GNN Expressiveness](https://mlsanity.com/paper/2304.07702/) (2023)
- [Directional Message Passing on Molecular Graphs via Synthetic Coordinates](https://mlsanity.com/paper/2111.04718/) (2021)
- [TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks](https://mlsanity.com/paper/2410.06530/) (2024)
- [Graph Sparsification via Mixture of Graphs](https://mlsanity.com/paper/2405.14260/) (2024)
- [Two-level Graph Neural Network](https://mlsanity.com/paper/2201.01190/) (2022)

## Built on by (most cited first, 1 of 1)

- [Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs](https://mlsanity.com/paper/2602.01454/) (2026)

## Abstract (arXiv)

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.