# A Topological characterisation of Weisfeiler-Leman equivalence classes

arXiv: https://arxiv.org/abs/2206.11876 · HTML: https://mlsanity.com/paper/2206.11876/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 23 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [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)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [The Surprising Power of Graph Neural Networks with Random Node Initialization](https://mlsanity.com/paper/2010.01179/) (2020)
- [Random Features Strengthen Graph Neural Networks](https://mlsanity.com/paper/2002.03155/) (2020)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Approximation Ratios of Graph Neural Networks for Combinatorial Problems](https://mlsanity.com/paper/1905.10261/) (2019)

## Built on by (most cited first, 3 of 3)

- [Heat Kernel Goes Topological](https://mlsanity.com/paper/2507.12380/) (2025)
- [Extending Graph Transformers with Quantum Computed Aggregation](https://mlsanity.com/paper/2210.10610/) (2022)
- [Weisfeiler-Lehman meets Events: An Expressivity Analysis for Continuous-Time Dynamic Graph Neural Networks](https://mlsanity.com/paper/2508.18052/) (2025)

## Abstract (arXiv)

Graph Neural Networks (GNNs) are learning models aimed at processing graphs and signals on graphs. The most popular and successful GNNs are based on message passing schemes. Such schemes inherently have limited expressive power when it comes to distinguishing two non-isomorphic graphs. In this article, we rely on the theory of covering spaces to fully characterize the classes of graphs that GNNs cannot distinguish. We then generate arbitrarily many non-isomorphic graphs that cannot be distinguished by GNNs, leading to the GraphCovers dataset. We also show that the number of indistinguishable graphs in our dataset grows super-exponentially with the number of nodes. Finally, we test the GraphCovers dataset on several GNN architectures, showing that none of them can distinguish any two graphs it contains.