On Halting vs Converging in Recurrent Graph Neural Networks

2026-04-28

AI Summary

This work investigates the impact of halting mechanisms and convergence behavior on the expressive power of recursive graph neural networks (RGNNs). Focusing on undirected graphs, it systematically compares fully convergent RGNNs, output-convergent RGNNs, and halting RGNNs equipped with vertex-wise halting classifiers, establishing for the first time an equivalence in expressive power between convergent and halting RGNN variants. To address desynchronization caused by asynchronous halting, the authors introduce a “traffic light” protocol, resolving an open problem posed by Bollen et al. Leveraging ReLU activations and sum aggregation, and employing tools from graded bisimulation invariance and monadic second-order logic (MSO), they prove that fully convergent RGNNs exactly capture graded modal μ-calculus (μGML), output-convergent RGNNs express at least μGML, and the model of Pflueger et al. retains full μGML expressiveness under guaranteed convergence.

Abstract

Recurrent Graph Neural Networks (RGNNs) extend standard GNNs by iterating message-passing until some stopping condition is met. Various RGNN models have been proposed in the literature. In this paper, we study three such models: converging RGNNs, where all vertex representations must stabilise; output-converging RGNNs, where only the output classifications must stabilise; and halting RGNNs, where a per-vertex halting classifier determines when to stop. We establish expressiveness relationships between these models: over undirected graphs, converging RGNNs are equally expressive as graded-bisimulation-invariant halting RGNNs, while output-converging RGNNs are at least as expressive. Combined with prior results on halting RGNNs, this shows that, relative to the classifiers expressible in monadic second-order logic (MSO), converging RGNNs express exactly the graded modal $μ$-calculus ($μ$GML), and output-converging RGNNs express at least $μ$GML. These results hold even when restricting to ReLU networks with sum aggregation. The main technical challenge is simulating halting RGNNs by converging ones: without a global halting classifier, vertices may locally decide to halt at different times, causing desynchronisation. We develop a "traffic-light" protocol that enables vertices to coordinate despite this asynchrony. Our results answer an open question from Bollen et al. (2025) and show that the RGNN model of Pflueger et al. (2024) retains full $μ$GML expressiveness even when convergence is guaranteed.

Problem

• Recurrent Graph Neural Networks
• expressiveness
• convergence
• halting
• graded bisimulation

Innovation

• Recurrent Graph Neural Networks
• expressiveness
• halting mechanism
• convergence
• traffic-light protocol