# How Hard Is It for Message-Passing GNNs to Simulate One Weisfeiler-Lehman Color-Refinement Step?

arXiv: https://arxiv.org/abs/2410.01308 · HTML: https://mlsanity.com/paper/2410.01308/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 2 Oct 2024.

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

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation](https://mlsanity.com/paper/2002.02126/) (2020)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Neural Graph Collaborative Filtering](https://mlsanity.com/paper/1905.08108/) (2019)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [Rethinking the Expressive Power of GNNs via Graph Biconnectivity](https://mlsanity.com/paper/2301.09505/) (2023)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Subgraph Neural Networks](https://mlsanity.com/paper/2006.10538/) (2020)
- [Reconstruction for Powerful Graph Representations](https://mlsanity.com/paper/2110.00577/) (2021)
- [Expressiveness and Approximation Properties of Graph Neural Networks](https://mlsanity.com/paper/2204.04661/) (2022)
- [From Relational Pooling to Subgraph GNNs: A Universal Framework for More Expressive Graph Neural Networks](https://mlsanity.com/paper/2305.04963/) (2023)
- [Extending the Design Space of Graph Neural Networks by Rethinking Folklore Weisfeiler-Lehman](https://mlsanity.com/paper/2306.03266/) (2023)
- [The Iteration Number of Colour Refinement](https://mlsanity.com/paper/2005.10182/) (2020)
- [Distance-Restricted Folklore Weisfeiler-Leman GNNs with Provable Cycle Counting Power](https://mlsanity.com/paper/2309.04941/) (2023)

## Built on by (most cited first, 2 of 2)

- [Your Vision-Language Model Can't Even Count to 20: Exposing the Failures of VLMs in Compositional Counting](https://mlsanity.com/paper/2510.04401/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)

## Abstract (arXiv)

Message-passing graph neural networks (MPGNNs) are commonly compared with the Weisfeiler-Lehman (WL) color-refinement procedure, but this comparison does not quantify the resource parameters a network needs to realize color refinement with bounded-size messages and finite numerical precision. We study the cost of simulating a single color-refinement step on unattributed graphs. We distinguish input-independent, or oblivious, simulation from instance-dependent simulation. In the former, the parameters, or their distributions in randomized models, are fixed before the input instance is known. Our results show that the local form of WL color refinement hides a global relabeling problem. In the oblivious setting, deterministic and zero-error randomized MPGNNs cannot solve this problem in the worst case using only shallow networks with small messages. We complement this lower bound with a nearly matching construction in a stronger rooted, port-aware model. By contrast, when the color set is large, bounded-error randomness can greatly reduce the cost, and a one-layer MPGNN with messages of logarithmic size and a logarithmic number of random bits suffices. We show that this logarithmic number of random bits is essentially necessary for shallow, small-message simulations. When the color set is small, we still obtain a rooted, port-aware simulation, but this construction requires more layers or larger messages. We also prove that this extra cost is partly unavoidable, as small color sets force a nontrivial trade-off between the number of layers and the message size. Finally, instance-dependent simulation can be much shallower, but the required instance-specific parameters are not necessarily easy to find. Together, these results reveal quantitative structure hidden behind the statement that MPGNNs match WL color refinement.