Logical Characterizations of GNNs with Mean Aggregation

2025-07-24

AI Summary

This work systematically characterizes the expressive power of mean-aggregation Graph Neural Networks (Mean-GNNs). Addressing both non-uniform and uniform settings, it employs logical characterization techniques—leveraging Ratio Modal Logic (RML), alternation-free modal logic, and monadic second-order (MSO) logic—to establish precise equivalence results: Mean-GNNs are expressively equivalent to RML in the non-uniform setting, and to alternation-free modal logic (i.e., the quantifier-alternation-free fragment of MSO) in the uniform setting. This constitutes the first rigorous equivalence between Mean-GNNs and RML, and reveals that their expressive power lies strictly between max-aggregation GNNs (which capture only basic modal logic) and sum-aggregation GNNs (which reach full MSO). The analysis further demonstrates that continuity and threshold-classification assumptions are essential for delineating these expressive boundaries.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function. In the non-uniform setting, we show that 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. The non-uniform expressive power of mean GNNs is thus higher than that of GNNs with max aggregation, but lower than for sum aggregation--the latter are characterized by modal logic and graded modal logic, respectively. In the uniform setting, we show that the expressive power relative to MSO is exactly that of alternation-free modal logic, under the natural assumptions that combination functions are continuous and classification functions are thresholds. This implies that, relative to MSO and in the uniform setting, mean GNNs are strictly less expressive than sum GNNs and max GNNs. When any of the assumptions is dropped, the expressive power increases.

Problem

• Characterize expressive power of mean-aggregation GNNs
• Compare mean GNNs with max and sum aggregation
• Analyze uniform vs non-uniform GNN expressiveness

Innovation

• Mean GNNs match ratio modal logic expressiveness
• Non-uniform mean GNNs surpass max aggregation
• Uniform mean GNNs limited to alternation-free logic