Towards Understanding the Expressive Power of GNNs with Global Readout

TL;DR

This paper investigates the expressive power of message-passing GNNs with global readout, showing they can capture certain first-order properties and identifying conditions under which their logical expressiveness aligns with graded modal logic.

Contribution

It demonstrates that sum aggregation and readout enable GNNs to express FO properties beyond C2 logic and identifies methods to restore C2 characterizability.

Findings

Sum aggregation and readout suffice for GNNs to capture FO properties beyond C2.

Limiting local aggregation or graph degree restores C2-like characterizability.

GNNs' expressive power exceeds C2 due to unbounded aggregation and readout interactions.

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 Wa{\l}{\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.