On Halting vs Converging in Recurrent Graph Neural Networks

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.

• The paper proves that converging RGNNs and halting RGNNs are expressively equivalent over undirected graphs using formal logic techniques.
• It introduces a novel traffic-light protocol to simulate asynchronous halting, ensuring local coordination and finite-time convergence under mild restrictions.
• The study links operational RGNN semantics to graded modal logic, providing practical insights for designing flexible and expressive graph neural networks.

Equivalence of Halting and Converging Semantics in Recurrent Graph Neural Networks

Introduction

"On Halting vs Converging in Recurrent Graph Neural Networks" (2604.25551) addresses fundamental questions regarding the expressiveness of Recurrent Graph Neural Networks (RGNNs) under different stopping semantics. The paper provides rigorous formalizations and comparisons for three output paradigms: converging RGNNs (where all vertex features must stabilize), output-converging RGNNs (where only output classifications stabilize), and halting RGNNs (with explicit per-vertex halting classifiers). The primary result is an expressiveness equivalence between converging and graded-bisimulation-invariant halting RGNNs over undirected graphs, subsuming models like those of Scarselli et al. and Pflueger et al. This equivalence is shown to persist under practical architectural choices, including ReLU-activated feedforward networks and sum aggregation.

Formal Definitions and Semantic Landscape

The paper recasts RGNN expressiveness through modal and monadic logics. GNN architectures, particularly aggregate-combine GNNs (AC-GNNs), are understood via their invariance under graded bisimulation. The authors leverage this connection to modal logics: fixed-depth GNNs correspond exactly to classifiers definable in graded modal logic, following Barceló et al. For RGNNs, expressiveness is less understood and is tightly linked to output semantics.

The three models are defined formally:

• Converging RGNNs: Require exact stabilization ($L(H_k) = H_k$) for all vertex features in finite time, without contraction or error thresholds.
• Output-converging RGNNs: Only the final output must stabilize; vertex features may oscillate if the output remains constant.
• Halting RGNNs: Each vertex halts when a halting classifier is satisfied. The output is read at the smallest step where all vertices halt.

Architectures are categorized as simple if their aggregation is summation and both update and readout functions are ReLU-activated feedforward networks.

Expressiveness Relations: Main Results

The central theorem asserts that, over undirected graphs, converging RGNNs and graded-bisimulation-invariant halting RGNNs are equally expressive. Strong claims are made:

• Equivalence: Any classifier expressible by a converging RGNN can be realized by a halting RGNN and vice versa (with mild restrictions).
• $\mu$GML Characterization: All classifiers definable in the graded modal $\mu$-calculus ($\mu$GML) are expressible by simple converging and output-converging RGNNs. This follows from canonical constructions and results from Bollen et al.

These results connect the practical termination protocols in RGNNs to logics with well-understood model-theoretic properties, yielding tight theoretical characterizations.

Converging-to-Halting Construction

Simulating a converging RGNN with a halting RGNN is achieved by duplicating vertex feature vectors (storing current and previous state). Stabilization is detected by local $\ell_1$ distance, triggering halting. The construction preserves simplicity under ReLU architectures.

Halting-to-Converging Construction

Simulating a halting RGNN with a converging RGNN is technically nontrivial due to asynchronous halting and desynchronization. The authors introduce a traffic-light protocol for local coordination: each vertex stores its current and previous snapshot, along with encoded traffic-light signals (one-hot vectors). Advancement occurs when vertices are eager (either they are behind or not yet halted) and aligned (they receive appropriate states from neighbors). Under mild restrictions (range and bounded-change conditions on halting classifier), this protocol can be implemented with simple AC-layers. The proof carefully manages asynchronous coordination, establishing convergence in finite time for all vertices in any connected component.

Logical Characterization and Open Questions

The work refines logical boundaries of RGNN expressiveness. Invariant under graded bisimulation, converging and output-converging RGNNs align with $\mu$GML within MSO-definable classifiers. Extending the Janin-Walukiewicz theorem to graded bisimulation, as conjectured, would seal this equivalence more tightly.

A noted limitation is that classifiers invariant under graded bisimulation but not MSO-definable (e.g., parity conditions on neighbor colors) are expressible, but their complete logical characterization remains open.

Practical and Theoretical Implications

The equivalence formalizes the operational semantics for RGNNs, clarifying that convergence-based and global halting protocols can simulate each other with no loss of expressiveness. This result informs the design of RGNNs for algorithmic tasks where precise logical characterization is required.

For practitioners, the findings allow flexibility in protocol design: using explicit halting mechanisms or waiting for stable configurations achieves the same class of classifiers. For theorists, the results motivate further exploration on expressive boundaries, aggregation function effects, and directed graph complications.

Future Developments

Several open questions remain:

• Directed Graphs: The equivalence proof leverages undirected graph structure. For directed graphs, local desynchronization and communication restrictions complicate convergence simulation.
• Aggregation Functions: The simplicity proof depends on summation or componentwise maxima for traffic signals. Mean aggregation introduces rational values, breaking the thresholding step for Boolean logic encoding.
• Logical Boundaries: Complete characterization beyond the MSO-definable fragment is unresolved. Visiting-acceptance semantics (Ahvonen et al.) yield expressive power orthogonal to $\mu$GML.

Resolving these gaps will further refine the interplay of GNN semantics and logic.

Conclusion

This paper delivers a rigorous, technically deep equivalence between halting and converging semantics in RGNNs, contingent on graded-bisimulation invariance over undirected graphs. The constructions are robust under practical architectural restrictions and directly tie RGNN expressiveness to modal logic. The work clarifies operational choices for RGNN design and charts new directions for theoretical explorations on expressive boundaries, protocol design, and logical characterization in graph learning.