# Weisfeiler and Lehman Go Paths: Learning Topological Features via Path Complexes

arXiv: https://arxiv.org/abs/2308.06838 · HTML: https://mlsanity.com/paper/2308.06838/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 54 references that are in the map.
- First posted on arXiv 13 Aug 2023.

## Builds on (most cited first, 30 of 33)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (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)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [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)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Directional Graph Networks](https://mlsanity.com/paper/2010.02863/) (2020)
- [UniGNN: a Unified Framework for Graph and Hypergraph Neural Networks](https://mlsanity.com/paper/2105.00956/) (2021)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Propagation Kernels](https://mlsanity.com/paper/1410.3314/) (2014)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Hierarchical Inter-Message Passing for Learning on Molecular Graphs](https://mlsanity.com/paper/2006.12179/) (2020)
- [A Short Tutorial on The Weisfeiler-Lehman Test And Its Variants](https://mlsanity.com/paper/2201.07083/) (2022)

## Built on by (most cited first, 11 of 11)

- [The Expressive Power of Graph Neural Networks: A Survey](https://mlsanity.com/paper/2308.08235/) (2023)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Cross-Domain Graph Data Scaling: A Showcase with Diffusion Models](https://mlsanity.com/paper/2406.01899/) (2024)
- [Higher-Order Topological Directionality and Directed Simplicial Neural Networks](https://mlsanity.com/paper/2409.08389/) (2024)
- [Multi-Scale Harmonic Encoding for Feature-Wise Graph Message Passing](https://mlsanity.com/paper/2505.15015/) (2025)
- [Rule Based Learning with Dynamic (Graph) Neural Networks](https://mlsanity.com/paper/2406.09954/) (2024)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Explainable Graph Representation Learning via Graph Pattern Analysis](https://mlsanity.com/paper/2512.04530/) (2025)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [Invariant-Based Weight Sharing for Message Passing](https://mlsanity.com/paper/2605.25750/) (2026)
- [Path Laplacian Encodings for Directed Graphs](https://mlsanity.com/paper/2610.04657/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs), despite achieving remarkable performance across different tasks, are theoretically bounded by the 1-Weisfeiler-Lehman test, resulting in limitations in terms of graph expressivity. Even though prior works on topological higher-order GNNs overcome that boundary, these models often depend on assumptions about sub-structures of graphs. Specifically, topological GNNs leverage the prevalence of cliques, cycles, and rings to enhance the message-passing procedure. Our study presents a novel perspective by focusing on simple paths within graphs during the topological message-passing process, thus liberating the model from restrictive inductive biases. We prove that by lifting graphs to path complexes, our model can generalize the existing works on topology while inheriting several theoretical results on simplicial complexes and regular cell complexes. Without making prior assumptions about graph sub-structures, our method outperforms earlier works in other topological domains and achieves state-of-the-art results on various benchmarks.