# Constant-Depth Rational-Weight GNNs and their Connection to Schanuel's Conjecture

arXiv: https://arxiv.org/abs/2610.16993 · HTML: https://mlsanity.com/paper/2610.16993/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 15 of 15)

- [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)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [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)
- [The Surprising Power of Graph Neural Networks with Random Node Initialization](https://mlsanity.com/paper/2010.01179/) (2020)
- [The Logic of Graph Neural Networks](https://mlsanity.com/paper/2104.14624/) (2021)
- [Graph Neural Networks with Local Graph Parameters](https://mlsanity.com/paper/2106.06707/) (2021)
- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks](https://mlsanity.com/paper/2211.03232/) (2022)
- [DNN Verification, Reachability, and the Exponential Function Problem](https://mlsanity.com/paper/2305.06064/) (2023)
- [On the power of graph neural networks and the role of the activation function](https://mlsanity.com/paper/2307.04661/) (2023)
- [On dimensionality of feature vectors in MPNNs](https://mlsanity.com/paper/2402.03966/) (2024)
- [Complexity of Neural Network Training and ETR: Extensions with Effectively Continuous Functions](https://mlsanity.com/paper/2305.11833/) (2023)
- [Repetition Makes Perfect: Recurrent Graph Neural Networks Match Message-Passing Limit](https://mlsanity.com/paper/2505.00291/) (2025)

## Abstract (arXiv)

A landmark result states that the expressive power of graph neural networks (GNNs) is bounded from above by the 1-dimensional Weisfeiler-Leman (1-WL) test. We ask whether a constant-depth GNN can match the expressive power of 1-WL while using only rational weights. We prove that this is the case when Schanuel's conjecture, a famous open problem from transcendental number theory, is taken to be true. Our GNN construction works even with integer weights, and more generally with a broad range of rational weights. As the activation function, it may use logistic sigmoid, hyperbolic tangent, or related functions. It is uniform across all graph sizes. We also prove a partial converse: if a GNN with exponentiation, hyperbolic sine or hyperbolic cosine as activation matches the expressive power of the 1-WL test in a slightly more expressive setting where graphs may contain certain auxiliary vertices, then this implies Schanuel's conjecture restricted to the numbers that arise in computations of the GNN. We also conduct experiments showing that the constructed GNNs indeed match the 1-WL test.