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

In this paper we cast neural networks defined on graphs as message-passing neural networks (MPNNs) to study the distinguishing power of different classes of such models. We are interested in when certain architectures are able to tell vertices apart based on the feature labels given as input with the graph. We consider two variants of MPNNS: anonymous MPNNs whose message functions depend only on the labels of vertices involved; and degree-aware MPNNs whose message functions can additionally use information regarding the degree of vertices. The former class covers popular graph neural network (GNN) formalisms for which the distinguished power is known. The latter covers graph convolutional networks (GCNs), introduced by Kipf and Welling, for which the distinguishing power was unknown. We obtain lower and upper bounds on the distinguishing power of (anonymous and degree-aware) MPNNs in terms of the distinguishing power of the Weisfeiler-Lehman (WL) algorithm. Our main results imply that (i) the distinguishing power of GCNs is bounded by the WL algorithm, but they may be one step ahead; (ii) the WL algorithm cannot be simulated by ``plain vanilla'' GCNs but the addition of a trade-off parameter between features of the vertex and those of its neighbours (as proposed by Kipf and Welling) resolves this problem.