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Logical Characterizations of GNNs with Mean Aggregation

Moritz Schönherr, Carsten Lutz

TL;DR

The paper investigates how the aggregation choice in GNNs (mean vs. sum vs. max) shapes their expressive power when viewed as vertex classifiers, across non-uniform and uniform settings relative to logical formalisms. It establishes a clear hierarchy: mean-based GNNs align with ratio modal logic non-uniformly and with modal logic uniformly (and thus match max-GNNs in some regimes), while mean-GNNs under natural continuity and thresholding collapse to alternation-free modal logic relative to MSO in the uniform setting. Through translations between logics and GNNs and by employing Ehrenfeucht–Fraïssé style games, it shows the practical insufficiency of naive mean-based constructions for expressive tasks, guiding the design of mean-aggregation GNNs. The work also contrasts these findings with sum- and max-aggregation networks, providing both theoretical insight and guidance for when mean aggregation is appropriate in large-scale graphs.

Abstract

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function, with the following results. In the non-uniform setting, such GNNs have exactly the same expressive power as ratio modal logic, which has modal operators expressing that at least a certain ratio of the successors of a vertex satisfies a specified property. In the uniform setting, the expressive power relative to MSO is exactly that of modal logic, and thus identical to the (absolute) expressive power of GNNs with max aggregation. The proof, however, depends on constructions that are not satisfactory from a practical perspective. This leads us to making the natural assumptions that combination functions are continuous and classification functions are thresholds. The resulting class of GNNs with mean aggregation turns out to be much less expressive: relative to MSO and in the uniform setting, it has the same expressive power as alternation-free modal logic. This is in contrast to the expressive power of GNNs with max and sum aggregation, which is not affected by these assumptions.

Logical Characterizations of GNNs with Mean Aggregation

TL;DR

The paper investigates how the aggregation choice in GNNs (mean vs. sum vs. max) shapes their expressive power when viewed as vertex classifiers, across non-uniform and uniform settings relative to logical formalisms. It establishes a clear hierarchy: mean-based GNNs align with ratio modal logic non-uniformly and with modal logic uniformly (and thus match max-GNNs in some regimes), while mean-GNNs under natural continuity and thresholding collapse to alternation-free modal logic relative to MSO in the uniform setting. Through translations between logics and GNNs and by employing Ehrenfeucht–Fraïssé style games, it shows the practical insufficiency of naive mean-based constructions for expressive tasks, guiding the design of mean-aggregation GNNs. The work also contrasts these findings with sum- and max-aggregation networks, providing both theoretical insight and guidance for when mean aggregation is appropriate in large-scale graphs.

Abstract

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function, with the following results. In the non-uniform setting, such GNNs have exactly the same expressive power as ratio modal logic, which has modal operators expressing that at least a certain ratio of the successors of a vertex satisfies a specified property. In the uniform setting, the expressive power relative to MSO is exactly that of modal logic, and thus identical to the (absolute) expressive power of GNNs with max aggregation. The proof, however, depends on constructions that are not satisfactory from a practical perspective. This leads us to making the natural assumptions that combination functions are continuous and classification functions are thresholds. The resulting class of GNNs with mean aggregation turns out to be much less expressive: relative to MSO and in the uniform setting, it has the same expressive power as alternation-free modal logic. This is in contrast to the expressive power of GNNs with max and sum aggregation, which is not affected by these assumptions.

Paper Structure

This paper contains 21 sections, 63 theorems, 113 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Preliminaries
  4. Graph Neural Networks.
  5. Notions of Expressive Power.
  6. Non-Uniform Setting
  7. Uniform Setting
  8. Uniform Setting, Reloaded
  9. Conclusion
  10. Proofs for Section \ref{['sect:prelims']}
  11. Proofs for Section \ref{['sect:nonuniform']}
  12. Proof of Theorem \ref{['lemma: rml-to-gnn-translation-nonuniform']}
  13. Proof of Theorem \ref{['theorem: rml-to-gnn-translation-nonuniform-univ-approx']}
  14. Proof of Theorems \ref{['lem:MLtomaxGNN']}, \ref{['lemma:\n mean-gnn-to-rml-non-uniform-translation']}, and \ref{['lem:frommaxgnntoML']}
  15. Proofs for Section \ref{['sect:uniform']}
  16. ...and 6 more sections

Key Result

Lemma 1

Let $\mathsf{AGG} \in \{ \mathsf{Max},\mathsf{Sum},\mathsf{Mean}\}$. If a property is definable by a simple$\mathsf{AGG}$-gnn with $\mathsf{ReLU}^*$ activation, then it is definable by a simple$\mathsf{AGG}$-gnn with $\mathsf{ReLU}$ activation.

Theorems & Definitions (101)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 91 more