How Expressive Are Graph Neural Networks in the Presence of Node Identifiers?

2026-01-29

AI Summary

This work investigates the expressive power of graph neural networks (GNNs) for purely structural node queries when nodes are augmented with unique identifiers. Drawing inspiration from order-invariant definability in finite model theory, the paper introduces the novel notion of "key-invariant expressiveness" to systematically characterize how unique identifiers influence GNN expressivity. By integrating modal logic with bounded-variable logics equipped with counting quantifiers, the study analyzes classes of GNNs employing local max or sum-based aggregation schemes, precisely delineating their expressive boundaries in the presence of unique identifiers. The analysis reveals how such identifiers enhance the ability of GNNs to capture structural properties of graphs that would otherwise be indistinguishable under standard message-passing architectures.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Web query analysis, representation and understanding

Abstract

Graph neural networks (GNNs) are a widely used class of machine learning models for graph-structured data, based on local aggregation over neighbors. GNNs have close connections to logic. In particular, their expressive power is linked to that of modal logics and bounded-variable logics with counting. In many practical scenarios, graphs processed by GNNs have node features that act as unique identifiers. In this work, we study how such identifiers affect the expressive power of GNNs. We initiate a study of the key-invariant expressive power of GNNs, inspired by the notion of order-invariant definability in finite model theory: which node queries that depend only on the underlying graph structure can GNNs express on graphs with unique node identifiers? We provide answers for various classes of GNNs with local max- or sum-aggregation.

Problem

• Graph Neural Networks
• Expressive Power
• Node Identifiers
• Graph Structure
• Finite Model Theory

Innovation

• graph neural networks
• expressive power
• node identifiers
• key-invariant
• finite model theory