A Logical View of GNN-Style Computation and the Role of Activation Functions

TL;DR

This paper analyzes the expressive power of graph neural networks (GNNs) through a logical framework, revealing that ReLU activations enable greater expressiveness than bounded, eventually constant activations, especially with linear layers.

Contribution

It introduces MPLang, a logical language for GNNs, and proves the first separation in expressiveness between unbounded ReLU and bounded activations with linear layers.

Findings

ReLU-based GNNs are strictly more expressive than those with bounded, eventually constant activations.

MPLang characterizes GNN expressiveness in terms of walk-summed features.

Unbounded ReLU activations enable more complex numerical queries than bounded activations.

Abstract

We study the numerical and Boolean expressiveness of MPLang, a declarative language that captures the computation of graph neural networks (GNNs) through linear message passing and activation functions. We begin with A-MPLang, the fragment without activation functions, and give a characterization of its expressive power in terms of walk-summed features. For bounded activation functions, we show that (under mild conditions) all eventually constant activations yield the same expressive power - numerical and Boolean - and that it subsumes previously established logics for GNNs with eventually constant activation functions but without linear layers. Finally, we prove the first expressive separation between unbounded and bounded activations in the presence of linear layers: MPLang with ReLU is strictly more powerful for numerical queries than MPLang with eventually constant activation functions, e.g., truncated ReLU. This hinges on subtle interactions between linear aggregation and eventually constant non-linearities, and it establishes that GNNs using ReLU are more expressive than those restricted to eventually constant activations and linear layers.