Logical Characterizations of GNNs with Mean Aggregation

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.