# Cooperative Sheaf Neural Networks

arXiv: https://arxiv.org/abs/2507.00647 · HTML: https://mlsanity.com/paper/2507.00647/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 52 references that are in the map.
- First posted on arXiv 1 Jul 2025.

## Builds on (most cited first, 30 of 40)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Where Did the Gap Go? Reassessing the Long-Range Graph Benchmark](https://mlsanity.com/paper/2309.00367/) (2023)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)

## Built on by (most cited first, 12 of 12)

- [On the Necessity of Learnable Sheaf Laplacians](https://mlsanity.com/paper/2603.05395/) (2026)
- [A Gauge Theory of Superposition: Toward a Sheaf-Theoretic Atlas of Neural Representations](https://mlsanity.com/paper/2603.00824/) (2026)
- [Learning Multi-Agent Coordination via Sheaf-ADMM](https://mlsanity.com/paper/2605.31005/) (2026)
- [Deep Neural Sheaf Diffusion](https://mlsanity.com/paper/2605.19021/) (2026)
- [Temporal Sheaf Neural Networks with Dynamic Orthogonal Transport](https://mlsanity.com/paper/2606.10071/) (2026)
- [SheafStain: Sheaf-Theoretic Schr"odinger Bridge for Spatially and Biologically Coherent Virtual Staining](https://mlsanity.com/paper/2606.11846/) (2026)
- [Hierarchical Pooling for Sheaf Neural Networks](https://mlsanity.com/paper/2606.20932/) (2026)
- [Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs](https://mlsanity.com/paper/2607.25387/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [Let the Heads Talk: Beyond Diagonal Graph Attention](https://mlsanity.com/paper/2610.01494/) (2026)
- [Mitigating Over-squashing without Rewiring: A Sheaf Effective Resistance Perspective](https://mlsanity.com/paper/2610.04157/) (2026)
- [PATTERN Can Be Solved from Memory: Generator-Aware Baselines for Synthetic Node Classification](https://mlsanity.com/paper/2610.15020/) (2026)

## Abstract (arXiv)

Sheaf diffusion has recently emerged as a promising design pattern for graph representation learning due to its inherent ability to handle heterophilic data and avoid oversmoothing. Meanwhile, cooperative message passing has also been proposed as a way to enhance the flexibility of information diffusion by allowing nodes to independently choose whether to propagate/gather information from/to neighbors. A natural question ensues: is sheaf diffusion capable of exhibiting this cooperative behavior? Here, we provide a negative answer to this question. In particular, we show that existing sheaf diffusion methods fail to achieve cooperative behavior due to the lack of message directionality. To circumvent this limitation, we introduce the notion of cellular sheaves over directed graphs and characterize their in- and out-degree Laplacians. We leverage our construction to propose Cooperative Sheaf Neural Networks (CSNNs). Theoretically, we characterize the receptive field of CSNN and show it allows nodes to selectively attend (listen) to arbitrarily far nodes while ignoring all others in their path, potentially mitigating oversquashing. Our experiments show that CSNN presents overall better performance compared to prior art on sheaf diffusion as well as cooperative graph neural networks.