# Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study

arXiv: https://arxiv.org/abs/2607.19514 · HTML: https://mlsanity.com/paper/2607.19514/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 21 Jul 2026.

## Builds on (most cited first, 12 of 12)

- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [The Surprising Power of Graph Neural Networks with Random Node Initialization](https://mlsanity.com/paper/2010.01179/) (2020)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [TopoBench: A Framework for Benchmarking Topological Deep Learning](https://mlsanity.com/paper/2406.06642/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Gauge-invariant representation holonomy](https://mlsanity.com/paper/2601.21653/) (2026)
- [On the Necessity of Learnable Sheaf Laplacians](https://mlsanity.com/paper/2603.05395/) (2026)
- [GraphUniverse: Synthetic Graph Generation for Evaluating Inductive Generalization](https://mlsanity.com/paper/2509.21097/) (2025)

## Abstract (arXiv)

Geometric architectures are often motivated by internal mechanisms, but accuracy alone does not show whether predictions use them. In Sheaf Neural Networks (SNNs), edge transports form a connection whose cycle products define holonomy. We ask whether training changes triangle holonomy, whether predictions rely on the learned connection, and whether holonomy drives triangle counting. We use basis-independent loop readouts with identity interventions and shortcut controls. On high-homophily GraphUniverse graphs, triangle counting increases the mean SO(2) triangle rotation in Neural Sheaf Propagation (NSP) from 0.010 to 0.388 radians, while community detection ends at 0.029 radians. With more data, learned SO(2)--NSP outperforms Identity NSP, and replacing its transports after training increases error further. However, ridge regression is more accurate, diagonal maps improve without continuous rotation, and fixed-degree models develop rotation without improved counting. Thus, NSP can learn and rely on a nontrivial connection, but our experiments do not show that triangle holonomy drives its predictions.