Lost in Aggregation: On a Fundamental Expressivity Limit of Message-Passing Graph Neural Networks

TL;DR

This paper demonstrates a fundamental limitation of message-passing graph neural networks (MP-GNNs), showing they cannot distinguish between many non-isomorphic graphs due to their limited expressive power, even with multiple iterations.

Contribution

The paper proves that MP-GNNs induce only polynomially many equivalence classes, revealing an inherent expressivity limit compared to the doubly-exponential number of non-isomorphic graphs.

Findings

MP-GNNs induce polynomially many equivalence classes

Two iterations of Color Refinement induce exponential classes

MP-GNNs are weaker than full graph isomorphism tests

Abstract

We define a generic class of functions that captures most conceivable aggregations for Message-Passing Graph Neural Networks (MP-GNNs), and prove that any MP-GNN model with such aggregations induces only a polynomial number of equivalence classes on all graphs - while the number of non-isomorphic graphs is doubly-exponential (in number of vertices). Adding a familiar perspective, we observe that merely 2-iterations of Color Refinement (CR) induce at least an exponential number of equivalence classes, making the aforementioned MP-GNNs relatively infinitely weaker. Previous results state that MP-GNNs match full CR, however they concern a weak, 'non-uniform', notion of distinguishing-power where each graph size may required a different MP-GNN to distinguish graphs up to that size. Our results concern both distinguishing between non-equivariant vertices and distinguishing between non-isomorphic graphs.