  We investigate the power of message-passing neural networks (MPNNs) in their
capacity to transform the numerical features stored in the nodes of their input
graphs. Our focus is on global expressive power, uniformly over all input
graphs, or over graphs of bounded degree with features from a bounded domain.
Accordingly, we introduce the notion of a global feature map transformer
(GFMT). As a yardstick for expressiveness, we use a basic language for GFMTs,
which we call MPLang. Every MPNN can be expressed in MPLang, and our results
clarify to which extent the converse inclusion holds. We consider exact versus
approximate expressiveness; the use of arbitrary activation functions; and the
case where only the ReLU activation function is allowed.
