Towards Understanding the Expressive Power of GNNs with Global Readout

2026-04-23

AI Summary

This work investigates the expressive power of message-passing graph neural networks with global readout (ACR-GNNs) with respect to first-order logical properties, focusing on their relationship to two-variable counting logic (C²). By integrating a graded modal logic framework augmented with a global counting modality, the study establishes—for the first time—that sum aggregation combined with a global readout mechanism suffices to exceed the expressiveness of C². It further demonstrates that the interaction between unbounded aggregation and global readout is essential for surpassing C²’s limitations. The paper also identifies two natural conditions under which C² characterizability is recovered, thereby delineating tight upper and lower bounds on the expressive capacity of ACR-GNNs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the expressive power of message-passing aggregate-combine-readout graph neural networks (ACR-GNNs). Particularly, we focus on the first-order (FO) properties expressible by this formalism. While a tight logical characterisation remains a difficult open question, we make two contributions towards answering it. First, we show that sum aggregation and readout suffice for GNNs to capture FO properties that cannot be expressed in the logic C2 on both directed and undirected graphs. This strengthens known results by Hauke and Wał{\k e}ga (2026) where aggregation and readout functions are specially crafted for the task. Second, we identify two natural ways of restoring characterisability (with regard to C2) for ACR-GNNs. One option is to limit local aggregation (without imposing restrictions on global readout), whilst the second is to run ACR-GNNs over graphs of bounded degree (but unbounded size). In both cases, the FO properties captured by GNNs are exactly those definable by a formula in graded modal logic with global counting modalities. Our results thus establish an innate lower- and upper-bound in terms of how far (fragments of) C2 can be taken to characterise GNNs, and imply that is indeed the unbounded interaction of aggregation and readout that pushes the logical expressive power of GNNs above C2.

Problem

• expressive power
• graph neural networks
• first-order logic
• C2 logic
• global readout

Innovation

• expressive power
• graph neural networks
• global readout
• first-order logic
• C2 logic