Structural Preservation and the Logical Expressiveness of Graph Neural Networks

Abstract: Bridges between graph neural networks (GNNs) and logical formalisms have been established by fixing architectural choices, such as the types of aggregation, combination, and activation functions. These choices define restricted classes of GNNs for which tight correspondences with logical formalisms can be obtained, by showing that logical formulae can be translated into equivalent GNNs and, conversely, that GNNs can be translated into equivalent formulae. In this paper we take a semantic perspective by establishing the logical expressiveness of classes of GNN classifiers that are preserved under structural properties: embeddings (extensions), injective homomorphisms, and homomorphisms. We show that, for each such property, there exists a fragment of graded modal logic characterising the class of GNNs. In particular, preservation under embeddings, injective homomorphisms, and homomorphisms corresponds to existential graded modal logic, its existential-positive fragment, and existential-positive modal logic, respectively. These results characterise the expressiveness of broad classes of GNNs independently of specific architectural choices, but we also show that each of these classes admits a GNN architecture of the same expressiveness. Technically, our approach uses a new well-quasi-order result for trees of bounded height, yielding finite representations of unravelling-invariant classes.

• The paper demonstrates that preservation under graph mappings corresponds to definability in specific modal logic fragments, ensuring finite logical characterizations.
• It introduces novel well-quasi-order results for bounded-height rooted trees to establish a finite set of minimal representatives for GNN invariants.
• Architectural classifications reveal that monotonic and MAX-aggregation GNNs capture distinct expressiveness limits, guiding robust model designs.

Structural Preservation and Logical Expressiveness in GNNs

Introduction

"Structural Preservation and the Logical Expressiveness of Graph Neural Networks" (2606.17882) presents a systematic and technically rigorous account of how structural preservation properties shape the logical expressiveness of GNN classifiers. The work departs from architecture-specific analyses and focuses on semantic preservation under graph mappings: embeddings, injective homomorphisms, and homomorphisms. The authors establish equivalences between these preservation notions and definability in graded modal logic fragments, offering new well-quasi-order (wqo) results to ensure finite characterizations. The paper also delivers tight architectural characterizations, determining precisely which GNN architectures correspond to these logical and semantic classes.

Semantic Framework and Preservation Notions

A key contribution is the adoption of a semantic, architecture-agnostic perspective, wherein the expressiveness of GNNs is analyzed via closure properties under graph mappings. Specifically, the paper considers invariance and preservation under:

• Embeddings: Structure-preserving injective graph homomorphisms onto induced subgraphs.
• Injective Homomorphisms: Injective structure-preserving mappings, allowing label monotonicity.
• Homomorphisms: Structure and label-preserving mappings, possibly many-to-one.

These notions transcend the conventional isomorphism-invariance typically imposed on GNNs. The semantic characterization aligns these invariance/preservation notions with fragments of graded modal logic as follows:

Preservation Notion	Logical Correspondence
Embedding	Existential Graded Modal Logic ($\mathbf{EML}$)
Injective Homomorphism	Existential-Positive Graded Modal Logic ($\mathbf{EPML}$)
Homomorphism	Existential-Positive Modal Logic ($\mathbf{EPML}^{=1}$)

This alignment enables a fine-grained analysis of the types of node properties GNNs can define or separate, abstracting over concrete choices of aggregation or activation.

Well-Quasi-Ordering and Finite Characterization

A significant technical advance is the derivation of new wqo results for rooted trees of bounded height under the aforementioned mappings. The use of locality (GNNs’ computation depending only on bounded-hop neighborhoods) allows reduction of expressiveness questions to classes of tree-unravelings.

The main theorem demonstrates that, for any structural relation among embedding, injective homomorphism, and homomorphism, classes of rooted trees of bounded height are well-quasi-ordered. This implies that every invariant under $L$-unraveling and preserved under a given mapping admits a finite set of minimal representatives, ensuring each class definable by a finite disjunction of characteristic modal formulas. This approach generalizes classic preservation theorems from finite model theory (e.g., Rossman’s homomorphism preservation, van Benthem–Rosen theorem), but extends them to GNN settings where existing preservation results do not straightforwardly apply.

Syntactic and Architectural Characterizations

The paper transitions from logical to architectural characterization by identifying GNN classes whose predictions are preserved under the corresponding graph mappings:

• Monotonic GNNs (MGNNs): GNNs where aggregate-combine functions are monotone (pointwise non-decreasing).
• $\mathrm{MAX}$-aggregation MGNNs: Monotone GNNs using $\mathrm{MAX}$ as the sole aggregation.
• Augmented MGNNs: MGNNs equipped with an initial pointwise augmentation layer (feature transformation).

Correspondences are established:

Architectural Class	Logical Expressiveness	Preservation Property
Augmented MGNNs	$\mathbf{EML}$	Embeddings
MGNNs	$\mathbf{EPML}$	Injective Homomorphisms
$\mathrm{MAX}$-MGNNs	$\mathbf{EPML}^{=1}$	Homomorphisms
Augmented $\mathbf{EPML}$0-MGNNs	$\mathbf{EPML}$1	Embeddings (for unary modal)

Notably, any MGNN preserved under injective homomorphisms (and thus, any monotonic GNN) can be implemented as a positive-weight GNN without expressive loss—a nontrivial result that demonstrates the sufficiency of non-negative parameter matrices for this subclass.

Logical Separation and Limitations

Expressiveness separation is sharply delineated:

• Augmented MGNNs can express arbitrary local properties, including non-monotone conditions on individual node features.
• MGNNs express only monotone, existential-positive properties—capable of encoding threshold and path-existence queries, but not parity or exact counts.
• $\mathbf{EPML}$2-MGNNs are limited to existential properties insensitive to multiplicity—thus they cannot express threshold or counting queries, only existence.
• Global properties (e.g., graph connectivity) and certain non-local or edge-based predicates are inherently out of scope due to both the locality imposed by GNN architecture and the logical fragments characterized.

Limitations include restriction to node-level classification, threshold-based predictors, and binary/discrete feature vectors. The approach does not encompass GNNs using global reasoning (e.g., attention pooling across the entire graph) or edge features, nor does it address aggregation operations outside the positive monotone family except via explicit separation results.

Practical and Theoretical Implications

This work provides a modular foundation for both verification of GNN-based models and design of architecture templates with predictable logical expressiveness. The logical characterizations underpin efforts in symbolic model checking, explainable GNN architectures, and neurosymbolic integration, as they enable transfer of model-theoretic invariance results to the GNN domain. Practically, the results motivate the use of syntactic monotonicity constraints and positive weights for applications requiring robustness to graph transformations (e.g., knowledge graph completion, rule-based inference). It also clarifies the tradeoff: increased structural preservation yields decreased expressiveness, which must be managed depending on the application.

The introduction of new wqo results for tree classes of bounded height opens directions for future research in generalization, architecture design, and graph logic, including the treatment of graph-level (rather than node-level) prediction and generalization to richer input spaces.

Conclusion

The paper presents a formal semantic and syntactic framework capturing the exact logical boundaries of GNN expressiveness under structural preservation properties. By leveraging new wqo results, the authors ensure that all such classes admit finite, logically explicit characterizations, leading to tight architectural correspondences. The analysis draws strong connections between finite model theory and neural graph representation learning, highlighting constraints and possibilities for future neuro-symbolic AI systems exploiting GNNs (2606.17882).