Towards Understanding the Expressive Power of GNNs with Global Readout

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.

• The paper demonstrates that sum-only ACR-GNNs achieve expressiveness beyond C2 limitations by capturing first-order properties like strict linear orders.
• It introduces a 6-layer ACR-GNN architecture that computes edge and path counts via sum aggregation, highlighting the advantage of global readout over local methods.
• The work establishes that bounding graph degree or aggregation multiplicity restricts expressiveness to graded modal logics, clarifying deployment limits for various graph types.

Understanding the Expressive Power of GNNs with Global Readout

Introduction and Motivation

The expressive power of message-passing graph neural networks (GNNs) is a foundational question for researchers exploiting these architectures in various domains. While prior work characterizes the logical expressiveness of basic GNN variants—primarily those with local aggregation steps—this paper, "Towards Understanding the Expressive Power of GNNs with Global Readout" (2604.22870), undertakes a formal investigation into Aggregate-Combine-Readout GNNs (ACR-GNNs). Here, the core focus is on how the use of sum-based aggregation and global readout interplay with the space of first-order (FO) properties, especially beyond those expressible in the two-variable first-order logic with counting quantifiers ($\mathrm{C}^2$).

While local GNNs are suitably characterized by fragments of graded modal logic, incorporating global readout (i.e., aggregation across all vertices irrespective of connectivity) provably increases the model's capacity relative to standard modal logics. The paper addresses previously unresolved questions: how aggregation and readout types influence expressiveness, whether standard (sum-based) versions can realize non-$\mathrm{C}^2$ properties, and the bounds that constrain this expressiveness for restricted graph domains.

Logical Characterization and Main Results

A principal contribution is the demonstration that simple ACR-GNNs using only unbounded sum operations for both aggregation and readout surpass the expressiveness of $\mathrm{C}^2$ formulas. Using a novel characterization of strict linear orders in terms of homomorphism counts, the authors construct FO properties not definable in $\mathrm{C}^2$ but still realizable by such GNNs. For instance, the set of directed graphs encoding (irreflexive, total, and transitive) strict linear orders—whose transitivity clause is inherently outside $\mathrm{C}^2$—is precisely encoded via GNNs computing edge and path counts, facilitated exclusively by sum aggregation/readout.

The argument extends to undirected graphs using careful gadgetization constructions, preserving the inexpressibility in $\mathrm{C}^2$ even after translating properties from directed to undirected settings. The critical novelty is that sum aggregation suffices, eschewing exotic or specifically tailored non-standard functions required in prior work, thereby strengthening existing lower bounds on ACR-GNN expressiveness.

Boundedness and Upper Bounds

Complementing the above, the study also explores means of restoring logical characterizability: specifically, by restricting the graph class to bounded degree, or by bounding the multiplicity-sensitivity of aggregation functions (making them $c$-bounded for some constant $c$). In both regimes, the expressive power of FO ACR-GNNs contracts to precisely that of graded modal logic with global counting modalities ($\mathrm{GML}^\exists$). In formal terms, the FO-definable vertex properties captured by ACR-GNNs in these settings are exactly those definable in $^\exists$—a robust fragment, but strictly less expressive than full FO for unrestricted graphs.

From a model-theoretic perspective, this is established via graded, $\mathrm{C}^2$0-turn bisimulations with global counting, demonstrating GNN invariance and thus matching expressiveness with modal logics extended by global modalities.

Technical Approach and Numerical Claims

The paper's technical centerpiece is a construction of a 6-layer simple ACR-GNN (using only sum and ReLU operations) that computes graph-wide sums relevant for characterizing strict linear orders via edge and path counts ($\mathrm{C}^2$1, $\mathrm{C}^2$2). The described construction avoids extraneous non-differentiable or hand-crafted logic—these quantities are computed straightforwardly through iterations of sum and linear layers, compatible with current GNN implementations.

Furthermore, the logical upper bounds for bounded-degree graphs are proved via homogeneity and type-based saturation, partitioning graphs into finitely many equivalence classes encoded via $\mathrm{C}^2$3 formulas. This tractability hinges crucially on the boundedness of degree or aggregation multiplicity, showing that—contrary to what might be expected—unbounded readout alone does not necessarily lift expressivity above modal logics unless paired with unbounded aggregation.

Strong claims include:

• Sum-only ACR-GNNs strictly surpass $\mathrm{C}^2$4 expressiveness, even without crafted aggregation/readout functions.
• Expressive equivalence to $\mathrm{C}^2$5 emerges exactly when aggregation is bounded or the graph degree is bounded.

Implications and Future Directions

The theoretical results have direct implications for both the deployment and analysis of GNNs in practical domains:

• For graph-structured data where node degree is naturally bounded (e.g., molecular graphs, road networks), the expressive envelope of GNNs is sharply characterized and does not exceed $\mathrm{C}^2$6, allowing principled reasoning about the limits of such deployments.
• For domains with unbounded degree (e.g., social networks), standard sum-based GNNs can capture FO-definable properties outside classical modal logics—including those defined by higher-order interaction patterns.
• The distinction between the role of aggregation and readout suggests potential directions for hybrid designs, where bounding only one (e.g., bounded aggregation, unbounded readout) may still restrict expressiveness, and could be exploited for inductive bias.

From a theoretical standpoint, the logical tightness results clarify the limitations of modal logics in characterizing GNN behavior, especially the necessity to go beyond $\mathrm{C}^2$7 and $\mathrm{C}^2$8 for real-world cases with unbounded relational complexity.

Unresolved questions remain regarding the precise demarcation for GNNs with unbounded sum aggregation but bounded readout, and how these insights transfer to more complex or heterogeneous GNN architectures, such as attention-based models.

Conclusion

This work systematically advances the formal understanding of ACR-GNN expressiveness, demonstrating that standard, sum-aggregation/readout GNNs can realize FO properties lying strictly beyond modal logics with bounded quantifier fragments, both for directed and undirected graphs. Simultaneously, it delineates clear boundaries on GNN expressiveness for bounded-degree graphs or aggregation, matching them with graded modal logics with global counting. These characterizations not only provide a rigorous differentiation between models in the GNN space but also set the stage for further research into the boundaries of expressivity, optimization of architectures for target logical fragments, and the quest for architectures aligning with practical graph constraints.