Logical Characterizations of GNNs with Mean Aggregation

TL;DR

This paper characterizes the expressive power of GNNs with mean aggregation, showing their limitations under practical assumptions and comparing them to GNNs with max and sum aggregations.

Contribution

It provides a logical characterization of GNNs with mean aggregation and analyzes their expressive power under realistic assumptions.

Findings

Mean aggregation GNNs are as expressive as ratio modal logic in non-uniform settings.

Under practical assumptions, mean aggregation GNNs match the expressive power of modal logic.

Max and sum aggregation GNNs retain their full expressive power regardless of assumptions.

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.