# Sheaf Neural Networks

arXiv: https://arxiv.org/abs/2012.06333 · HTML: https://mlsanity.com/paper/2012.06333/

## Key facts

- Cited by 84 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2020.

## Built on by (most cited first, 30 of 84)

- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Sheaf Hypergraph Networks](https://mlsanity.com/paper/2309.17116/) (2023)
- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module](https://mlsanity.com/paper/2305.16174/) (2023)
- [Simplicial Representation Learning with Neural k-Forms](https://mlsanity.com/paper/2312.08515/) (2023)
- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [Bundle Neural Networks for message diffusion on graphs](https://mlsanity.com/paper/2405.15540/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Cooperative Sheaf Neural Networks](https://mlsanity.com/paper/2507.00647/) (2025)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Generalized energy and gradient flow via graph framelets](https://mlsanity.com/paper/2210.04124/) (2022)
- [Prospects for inconsistency detection using large language models and sheaves](https://mlsanity.com/paper/2401.16713/) (2024)
- [Heterogeneous Sheaf Neural Networks](https://mlsanity.com/paper/2409.08036/) (2024)
- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [Joint Diffusion Processes as an Inductive Bias in Sheaf Neural Networks](https://mlsanity.com/paper/2407.20597/) (2024)
- [Polynomial Neural Sheaf Diffusion: A Spectral Filtering Approach on Cellular Sheaves](https://mlsanity.com/paper/2512.00242/) (2025)
- [Sheaf HyperNetworks for Personalized Federated Learning](https://mlsanity.com/paper/2405.20882/) (2024)
- [Bayesian Sheaf Neural Networks](https://mlsanity.com/paper/2410.09590/) (2024)
- [Nonlinear Sheaf Diffusion in Graph Neural Networks](https://mlsanity.com/paper/2403.00337/) (2024)
- [DeepSN: A Sheaf Neural Framework for Influence Maximization](https://mlsanity.com/paper/2412.12416/) (2024)
- [SIGMA: Sheaf-Informed Geometric Multi-Agent Pathfinding](https://mlsanity.com/paper/2502.06440/) (2025)
- [Sheaves Reloaded: A Directional Awakening](https://mlsanity.com/paper/2506.02842/) (2025)
- [Categorical Equivariant Deep Learning: Category-Equivariant Neural Networks and Universal Approximation Theorems](https://mlsanity.com/paper/2511.18417/) (2025)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Deep Learning and Geometric Deep Learning: an introduction for mathematicians and physicists](https://mlsanity.com/paper/2305.05601/) (2023)
- [Dirichlet Energy Enhancement of Graph Neural Networks by Framelet Augmentation](https://mlsanity.com/paper/2311.05767/) (2023)

## Abstract (arXiv)

We present a generalization of graph convolutional networks by generalizing the diffusion operation underlying this class of graph neural networks. These sheaf neural networks are based on the sheaf Laplacian, a generalization of the graph Laplacian that encodes additional relational structure parameterized by the underlying graph. The sheaf Laplacian and associated matrices provide an extended version of the diffusion operation in graph convolutional networks, providing a proper generalization for domains where relations between nodes are non-constant, asymmetric, and varying in dimension. We show that the resulting sheaf neural networks can outperform graph convolutional networks in domains where relations between nodes are asymmetric and signed.