# Gradual Weisfeiler-Leman: Slow and Steady Wins the Race

arXiv: https://arxiv.org/abs/2209.09048 · HTML: https://mlsanity.com/paper/2209.09048/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 37 references that are in the map.
- First posted on arXiv 19 Sep 2022.

## Builds on (most cited first, 18 of 18)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [A Survey on Graph Kernels](https://mlsanity.com/paper/1903.11835/) (2019)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Wasserstein Weisfeiler-Lehman Graph Kernels](https://mlsanity.com/paper/1906.01277/) (2019)
- [Propagation Kernels](https://mlsanity.com/paper/1410.3314/) (2014)
- [Graph Kernels: State-of-the-Art and Future Challenges](https://mlsanity.com/paper/2011.03854/) (2020)
- [On Valid Optimal Assignment Kernels and Applications to Graph Classification](https://mlsanity.com/paper/1606.01141/) (2016)
- [Faster Kernels for Graphs with Continuous Attributes via Hashing](https://mlsanity.com/paper/1610.00064/) (2016)
- [subgraph2vec: Learning Distributed Representations of Rooted Sub-graphs from Large Graphs](https://mlsanity.com/paper/1606.08928/) (2016)
- [The Power of the Weisfeiler-Leman Algorithm for Machine Learning with Graphs](https://mlsanity.com/paper/2105.05911/) (2021)
- [Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement](https://mlsanity.com/paper/1509.08251/) (2015)
- [The Iteration Number of Colour Refinement](https://mlsanity.com/paper/2005.10182/) (2020)
- [A Generalized Weisfeiler-Lehman Graph Kernel](https://mlsanity.com/paper/2101.08104/) (2021)
- [Learning subtree pattern importance for Weisfeiler-Lehmanbased graph kernels](https://mlsanity.com/paper/2106.04739/) (2021)
- [Wasserstein Graph Distance Based on L1-Approximated Tree Edit Distance between Weisfeiler-Lehman Subtrees](https://mlsanity.com/paper/2207.04216/) (2022)
- [Message Passing Graph Kernels](https://mlsanity.com/paper/1808.02510/) (2018)

## Built on by (most cited first, 5 of 5)

- [Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks](https://mlsanity.com/paper/2211.03232/) (2022)
- [Exploring Consistency in Graph Representations:from Graph Kernels to Graph Neural Networks](https://mlsanity.com/paper/2410.23748/) (2024)
- [Deep Hierarchical Graph Alignment Kernels](https://mlsanity.com/paper/2405.05545/) (2024)
- [Towards Subgraph Isomorphism Counting with Graph Kernels](https://mlsanity.com/paper/2405.07497/) (2024)
- [Caterpillar GNN: Replacing Message Passing with Efficient Aggregation](https://mlsanity.com/paper/2506.06784/) (2025)

## Abstract (arXiv)

The classical Weisfeiler-Leman algorithm aka color refinement is fundamental for graph learning with kernels and neural networks. Originally developed for graph isomorphism testing, the algorithm iteratively refines vertex colors. On many datasets, the stable coloring is reached after a few iterations and the optimal number of iterations for machine learning tasks is typically even lower. This suggests that the colors diverge too fast, defining a similarity that is too coarse. We generalize the concept of color refinement and propose a framework for gradual neighborhood refinement, which allows a slower convergence to the stable coloring and thus provides a more fine-grained refinement hierarchy and vertex similarity. We assign new colors by clustering vertex neighborhoods, replacing the original injective color assignment function. Our approach is used to derive new variants of existing graph kernels and to approximate the graph edit distance via optimal assignments regarding vertex similarity. We show that in both tasks, our method outperforms the original color refinement with only a moderate increase in running time advancing the state of the art.