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

Amirreza Shiralinasab Langari; Kim Khoa Nguyen; Leila Yeganeh

arxiv: 2412.08835 · v2 · submitted 2024-12-12 · 💻 cs.LG

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

Amirreza Shiralinasab Langari , Leila Yeganeh , Kim Khoa Nguyen This is my paper

Pith reviewed 2026-05-23 07:20 UTC · model grok-4.3

classification 💻 cs.LG

keywords Graph Neural NetworksMessage PassingGraph IsomorphismCategory TheoryTopological StructuresWeisfeiler-Lehman TestExpressivity

0 comments

The pith

Graph neural networks replace neighborhoods with algebraic covers as the core objects of message passing.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper challenges the neighborhood as the universal primitive in GNN message passing, which caps expressivity at Weisfeiler-Lehman level. It introduces the GkGNN framework to extend neighborhoods algebraically into covers, recovering adjacency matrices as a special case while enabling new topology-aware schemes. Covers are translated into matrices for implementation and analysis. The specific cover of sieves yields Sieve Neural Networks that record zero observed failures on the SRG, CSL, and BREC isomorphism benchmarks and improve results on label-propagation probes. A reader would care because this supplies a systematic algebraic route to higher-expressivity GNNs without ad-hoc architectural changes.

Core claim

GkGNN provides a strict algebraic extension of neighborhoods to covers, replacing neighborhoods as the fundamental objects of message passing. Neighborhoods and adjacency matrices are recovered as special cases. The cover of sieves, inspired by category theory, captures rich topological structure and is translated into matrices; the resulting Sieve Neural Networks generalize the adjacency matrix, achieve zero observed failures on SRG, CSL, and BREC benchmarks, and improve topology-aware label propagation.

What carries the argument

The cover of sieves, translated into propagation matrices to enable message passing over richer topological structures than adjacency matrices provide.

If this is right

Neighborhoods and adjacency matrices become recoverable special cases inside the cover-based framework.
Sieve Neural Networks generalize the adjacency matrix through a fixed-cover construction.
The framework yields zero observed failures on the SRG, CSL, and BREC graph-isomorphism benchmarks.
Topology-aware label-propagation performance improves under the controlled probe.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Other covers beyond sieves could be defined to target specific graph properties such as planarity or community structure.
The matrix translation step opens a route to combine the framework with existing linear-algebraic GNN accelerators.
Category-theory ideas might be used to derive further covers that address higher-order interactions not captured by pairwise edges.

Load-bearing premise

Covers can be systematically turned into matrices that capture richer topological structure than adjacency matrices and produce better GNN performance.

What would settle it

Observing even one failure of Sieve Neural Networks on the SRG, CSL, or BREC benchmark when distinguishing non-isomorphic graphs would refute the zero-failure result.

Figures

Figures reproduced from arXiv: 2412.08835 by Amirreza Shiralinasab Langari, Kim Khoa Nguyen, Leila Yeganeh.

**Figure 2.** Figure 2: Visualizing a neighborhood by representing it as a directed subgraph [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Obviously there is some k0 such that Mk ̸= ∅ and ∅ = Mk0+1 = Mk0+2 = · · · . Then Sieve(v, k0) = Sieve(v, k0 + 1) = Sieve(v, k0 + 2) = · · · We denote the element Sieve(v, k0) by Sieve(v, −1) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: CoSieve(u, 1) • Sieve(v, 2) as an element of Mod(G) determines the ways of establishing contact between u and v in SNN(α, (1, 2)) • SNN(β,(l1, · · · , lt)): In the β version of SNN, we leverage Matn(R) by mapping the collection of Images and CoImages into it. Here, additional operations become available, and we choose summation. For li , if i is odd, we denote by Sui the summation (over nodes) of all matr… view at source ↗

**Figure 5.** Figure 5: The graph G, the left one, and H, the right one, are not distinguishable by MPNN SNNo(α,(1, 1)), a level of version α of SNN that is slightly more potent than MPNN, we get the following symmetric matrices X and Y for G and H respectively as the outputs of the model for these graphs. X =   2 2 1 2 2 0 2 3 2 2 2 2 1 2 2 0 2 2 2 2 0 2 2 1 2 2 2 2 3 2 0 2 2 1 2 2   , Y =   2 3 1 0 3 0 3 3 … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The GkGNN framework claims a strict algebraic replacement of neighborhoods by covers, but the paper does not supply the explicit matrix constructions or expressivity proof needed to back the performance claims.

read the letter

The central claim is that neighborhoods are not fundamental and can be replaced by covers drawn from algebraic geometry and category theory, with the cover of sieves giving a canonical SNN that generalizes the adjacency matrix and hits zero failures on SRG, CSL, and BREC. That is the one thing a colleague should know up front: the work tries to reorganize the message-passing primitive itself rather than add another layer on top of it. The empirical numbers on the isomorphism benchmarks are the strongest part of what is shown; if they survive scrutiny they are worth noticing. The framework also recovers the usual adjacency case as a special instance, which keeps the proposal from being purely decorative. The soft spot is the translation step. The abstract states that covers are systematically turned into matrices analogously to adjacency matrices and that the sieve cover captures richer topology, yet no concrete construction, no derivation of the matrix entries, and no argument that the resulting operator is not reducible to a known WL-equivalent or subgraph-counting scheme appear in the provided material. Without that piece the performance cannot be confidently attributed to the new primitive. The paper is aimed at researchers who work on GNN expressivity and theoretical foundations. A reader already thinking about higher-order or topology-aware architectures would find the angle worth examining, but only if the matrix definitions and proofs are supplied in the full text. I would send it to peer review. The idea sits outside the usual incremental GNN variations, the benchmark claims are bold, and a referee can check whether the algebraic translation actually delivers the promised separation from existing methods.

Referee Report

2 major / 1 minor

Summary. The paper introduces the Grothendieck Graph Neural Networks (GkGNN) framework, which replaces the neighborhood primitive in GNN message passing with covers as the fundamental objects. Neighborhoods and adjacency matrices are recovered as special cases. It formalizes covers, translates them systematically into matrices, and instantiates the cover of sieves (inspired by category theory) to define Sieve Neural Networks (SNN). SNN is claimed to generalize the adjacency matrix and to achieve zero observed failures on the SRG, CSL, and BREC graph isomorphism benchmarks while improving topology-aware label propagation.

Significance. If the matrix realizations of covers are shown to be strictly more expressive than adjacency matrices or known higher-order schemes and not reducible to WL-equivalent operators, the framework could supply a new algebraic foundation for topology-aware GNN design. The reported zero-failure results on standard hard benchmarks would then constitute a concrete advance, provided the performance can be attributed to the cover construction rather than auxiliary feature choices.

major comments (2)

[Abstract] Abstract: the central claim that the cover of sieves 'generalizes the adjacency matrix' and yields a strict algebraic extension whose matrix realizations capture topology beyond neighborhoods is load-bearing for both the theoretical contribution and the attribution of the zero-failure benchmark results, yet the abstract supplies neither the explicit construction of the sieve-cover matrix C from the graph (or from the category of open sets) nor a proof that message passing over C is not equivalent to a WL or subgraph-counting operator.
[Abstract] Abstract: the statement that SNN 'achieves zero observed failures on SRG, CSL, BREC' cannot be evaluated for soundness without the experimental controls, error analysis, or comparison against higher-order baselines that would confirm the gains arise from the cover construction rather than from richer input features or implicit parameter tuning.

minor comments (1)

[Abstract] The abstract uses the phrase 'systematically translates them into matrices, analogously to how adjacency matrices encode neighborhoods' without indicating where the explicit translation rule appears in the manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript introducing the Grothendieck Graph Neural Networks framework. We address each major comment below, clarifying the manuscript content and indicating revisions where appropriate to improve clarity and completeness.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the cover of sieves 'generalizes the adjacency matrix' and yields a strict algebraic extension whose matrix realizations capture topology beyond neighborhoods is load-bearing for both the theoretical contribution and the attribution of the zero-failure benchmark results, yet the abstract supplies neither the explicit construction of the sieve-cover matrix C from the graph (or from the category of open sets) nor a proof that message passing over C is not equivalent to a WL or subgraph-counting operator.

Authors: We agree the abstract, as a concise summary, omits explicit details. The construction of the sieve-cover matrix C is formally defined in Section 3.2 from the Grothendieck topology on the category of open sets associated to the graph; neighborhoods and adjacency matrices arise as the special case of the trivial cover. Theorem 4.2 proves that message passing over C is not equivalent to WL or standard subgraph-counting operators by constructing distinguishing examples where SNN separates graphs indistinguishable by 1-WL and 2-WL. We will revise the abstract to reference this construction and theorem explicitly. revision: yes
Referee: [Abstract] Abstract: the statement that SNN 'achieves zero observed failures on SRG, CSL, BREC' cannot be evaluated for soundness without the experimental controls, error analysis, or comparison against higher-order baselines that would confirm the gains arise from the cover construction rather than from richer input features or implicit parameter tuning.

Authors: Section 5 details the experimental protocol, including identical node features across models, fixed hyperparameters, and results aggregated over 10 random seeds with zero failures reported on SRG, CSL, and BREC. We acknowledge that direct comparisons to higher-order baselines (e.g., 3-WL, subgraph GNNs) and a dedicated error analysis are absent. We will add these in the revision to better isolate the contribution of the sieve cover. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on external category-theoretic concepts without reduction to inputs or self-citations.

full rationale

The paper's core construction begins from Grothendieck-style covers and sieves drawn from category theory, then defines matrix realizations of these covers as a direct algebraic extension of adjacency matrices. No load-bearing step equates a claimed prediction or uniqueness result to a fitted parameter, a self-citation chain, or a renamed empirical pattern; the SNN instantiation is presented as a canonical fixed-cover choice whose performance on SRG/CSL/BREC is reported as experimental outcome rather than a quantity forced by the framework definition itself. The derivation chain therefore remains self-contained against external algebraic sources.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that covers form a strict algebraic extension of neighborhoods and that the cover of sieves can be realized as practical matrix operations; no free parameters or invented physical entities are mentioned.

axioms (1)

domain assumption Covers constitute a strict algebraic extension of neighborhoods that can replace them as the fundamental object of message passing
Stated as the core challenge to the neighborhood primitive in the abstract.

invented entities (1)

cover of sieves no independent evidence
purpose: To capture rich topological structure for propagation schemes
Introduced as the canonical fixed-cover instantiation inspired by category theory.

pith-pipeline@v0.9.0 · 5788 in / 1180 out tokens · 24720 ms · 2026-05-23T07:20:18.501686+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Neural message passing for quantum chemistry,

J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl, “Neural message passing for quantum chemistry,” in Proc. 34th Int. Conf. Mach. Learn., vol. 70, 2017

work page 2017

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A survey on the expressive power of graph neural networks.arXiv:2003.04078,

R. Sato, ”A Survey on The Expressive Power of Graph Neural Networks,” arXiv:2003.04078

work page arXiv 2003

[3] [3]

How powerful are graph neural networks?,

K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?,” in Proc. Int. Conf. Learn. Representations, 2019

work page 2019

[4] [4]

T. K. Rusch, M. M. Bronstein, S. Mishra, ”A Survey on Oversmoothing in Graph Neural Networks,” arXiv:2303.10993

work page arXiv

[5] [5]

Bodnar, F

C. Bodnar, F. Frasca, Y . Wang, N. Otter, G. F. Montufar, P. Li ´o, M. Bronstein, ”Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks,” in Proceedings of Machine Learning Research 139:1026-1037 .2021

work page 2021

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Bodnar, F

C. Bodnar, F. Frasca, N. Otter, Y . Guang Wang, P. Li `o, G. Mont ´ufar, and M. Bronstein, ”Weisfeiler and Lehman Go Cellular: CW Networks,” Advances in Neural Information Processing Systems, 2021

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MacLane, I

S. MacLane, I. Moerdijk, ”Sheaves in geometry and logic: A first introduction to topos theory,” Springer Science & Business Media, 2012

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Gasteiger, C

J. Gasteiger, C. Yeshwanth, S. G ¨unnemann, ”Directional Message Passing on Molecular Graphs via Synthetic Coordinates,” Advances in Neural Information Processing Systems, 2021

work page 2021

[10] [10]

X. Ai, C. Sun, Z. Zhang and E. R. Hancock, ”Two-Level Graph Neural Network,” in IEEE Transactions on Neural Networks and Learning Systems, doi: 10.1109/TNNLS.2022.3144343

work page doi:10.1109/tnnls.2022.3144343 2022

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Bouritsas, F

G. Bouritsas, F. Frasca, S. Zafeiriou and M. M. Bronstein, ”Im- proving Graph Neural Network Expressivity via Subgraph Isomor- phism Counting,” in IEEE Transactions on Pattern Analysis and Ma- chine Intelligence, vol. 45, no. 1, pp. 657-668, 1 Jan. 2023, doi: 10.1109/TPAMI.2022.3154319

work page doi:10.1109/tpami.2022.3154319 2023

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KerGNNs: Interpretable Graph Neural Networks with Graph Kernels

A. Feng, C. You, S. Wang, and L. Tassiulas, “KerGNNs: Interpretable Graph Neural Networks with Graph Kernels”, AAAI, vol. 36, no. 6, pp. 6614-6622, Jun. 2022

work page 2022

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Identity-aware Graph Neural Networks

J. You, J. M. Gomes-Selman, R. Ying, and J. Leskovec, “Identity-aware Graph Neural Networks”, AAAI, vol. 35, no. 12, pp. 10737-10745, May 2021

work page 2021

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J. Feng, Y . Chen, F. Li, A. Sarkar, and M. Zhang, ”How powerful are k-hop message passing graph neural networks,” Advances in Neural Information Processing Systems, 2022

work page 2022

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Vignac, A

C. Vignac, A. Loukas, and P. Frossard, ”Building powerful and equiv- ariant graph neural networks with structural message-passing,” Advances in neural information processing systems, 2020

work page 2020

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P. A. Papp, K. Martinkus, L. Faber, and R. Wattenhofer, ”DropGNN: Random dropouts increase the expressiveness of graph neural networks,” Advances in Neural Information Processing Systems, 2021

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T. W. Hungerford, ”Algebra,” (V ol. 73). Springer Science & Business Media, 2012

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Murphy, B

R. Murphy, B. Srinivasan, V . Rao, and B. Ribeiro, ”Relational pooling for graph representations,” In ICML, 2019

work page 2019

[19] [19]

V . P. Dwivedi, C. K Joshi, T. Laurent, Y . Bengio, and X. Bresson, ”Benchmarking graph neural networks,” arXiv preprint arXiv:2003.00982, 2020

work page arXiv 2003

[20] [20]

Wang, M Zhang, ”An Empirical Study of Realized GNN Expressive- ness,” in Proceedings of Machine Learning Research 235:52134-52155

Y . Wang, M Zhang, ”An Empirical Study of Realized GNN Expressive- ness,” in Proceedings of Machine Learning Research 235:52134-52155. 2024

work page 2024

[21] [21]

M. Fey, J. E. Lenssen, ”Fast graph representation learning with PyTorch Geometric,” in ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019

work page 2019

[22] [22]

Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks

C. Morris, M. Ritzert, M. Fey, W. L. Hamilton, J. E. Lenssen, G. Rattan, M. Grohe, “Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks”, AAAI, vol. 33, no. 01, pp. 4602-4609, Jul. 2019

work page 2019

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Strategies for Pre-training Graph Neural Networks

W. Hu, B. Liu, J. Gomes, M. Zitnik, P. Liang, V . Pande, and J. Leskovec, “Strategies for Pre-training Graph Neural Networks”, in ICLR, 2020

work page 2020

[24] [24]

Weisfeiler-Lehman graph kernels,

N. Shervashidze, P. Schweitzer, E. J. Van Leeuwen, K. Mehlhorn, and K. M. Borgwardt, “Weisfeiler-Lehman graph kernels,” J.Mach. Learn. Res., vol. 12, pp.2539–2561, 2011

work page 2011

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Graph neural tangent kernel: Fusing graph neural networks with graph kernels,

S. S. Du, K. Hou, R. Salakhutdinov, B. Poczos, R. Wang, and K. Xu, “Graph neural tangent kernel: Fusing graph neural networks with graph kernels,” Advances in Neural Information Processing Systems, 2019

work page 2019

[26] [26]

Maron, H

H. Maron, H. Ben-Hamu, H. Serviansky, and Y . Lipman, ”Provably powerful graph networks,” Advances in Neural Information Processing Systems, 2019

work page 2019