# Upper Bounds on the Quantifier Depth for Graph Differentiation in First-Order Logic

arXiv: https://arxiv.org/abs/1605.03480 · HTML: https://mlsanity.com/paper/1605.03480/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2016.

## Built on by (most cited first, 7 of 7)

- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [SpeqNets: Sparsity-aware Permutation-equivariant Graph Networks](https://mlsanity.com/paper/2203.13913/) (2022)
- [WL meet VC](https://mlsanity.com/paper/2301.11039/) (2023)
- [Fine-grained Expressivity of Graph Neural Networks](https://mlsanity.com/paper/2306.03698/) (2023)
- [The Power of the Weisfeiler-Leman Algorithm for Machine Learning with Graphs](https://mlsanity.com/paper/2105.05911/) (2021)
- [On the Expressiveness and Generalization of Hypergraph Neural Networks](https://mlsanity.com/paper/2303.05490/) (2023)
- [Homomorphism Expressivity of Spectral Invariant Graph Neural Networks](https://mlsanity.com/paper/2503.00485/) (2025)

## Abstract (arXiv)

We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is asymptotically not tight. In the logic setting, this translates to the statement that if two graphs of size n can be distinguished by a formula in first-order logic with counting with 3 variables (i.e., in C3), then they can also be distinguished by a C3-formula that has quantifier depth at most O(n^2/log(n)). To prove the result we define a game between two players that enables us to decouple the causal dependencies between the processes happening simultaneously over several iterations of the algorithm. This allows us to treat large color classes and small color classes separately. As part of our proof we show that for graphs with bounded color class size, the number of iterations until stabilization is at most linear in the number of vertices. This also yields a corresponding statement in first-order logic with counting. Similar results can be obtained for the respective logic without counting quantifiers, i.e., for the logic L3.