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

TL;DR

This paper investigates how the presence of unique node identifiers influences the expressive power of graph neural networks, linking their capabilities to logical definability and analyzing different GNN classes.

Contribution

It introduces a study of key-invariant expressive power of GNNs with node identifiers, connecting GNN expressiveness to finite model theory and logical invariance.

Findings

Characterizes the expressive power of GNNs with node identifiers.

Links GNN expressiveness to logical definability and invariance.

Provides insights into which node queries GNNs can express.

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.