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In this paper we cast neural networks defined on graphs as message-passingneural networks (MPNNs) in order to study the distinguishing power of differentclasses of such models. We are interested in whether certain architectures areable to tell vertices apart based on the feature labels given as input with thegraph. We consider two variants of MPNNS: anonymous MPNNs whose messagefunctions depend only on the labels of vertices involved; and degree-awareMPNNs in which message functions can additionally use information regarding thedegree of vertices. The former class covers a popular formalisms for computingfunctions on graphs: graph neural networks (GNN). The latter covers theso-called graph convolutional networks (GCNs), a recently introduced variant ofGNNs by Kipf and Welling. We obtain lower and upper bounds on thedistinguishing power of MPNNs in terms of the distinguishing power of theWeisfeiler-Lehman (WL) algorithm. Our results imply that (i) the distinguishingpower of GCNs is bounded by the WL algorithm, but that they are one step ahead;(ii) the WL algorithm cannot be simulated by "plain vanilla" GCNs but theaddition of a trade-off parameter between features of the vertex and those ofits neighbours (as proposed by Kipf and Welling themselves) resolves thisproblem.

Let's Agree to Degree: Comparing Graph Convolutional Networks in the Message-Passing Framework