The expressive power of Graph Neural Networks: a unifying point of view

Abstract

Graph Neural Networks (GNNs) are a wide class of connectionist models for graph processing. Recent studies have linked the expressive power of GNNs to the Weisfeiler– Lehman algorithm, which is a method of verifying whether two graphs are isomorphic or not. On the other hand, it was also observed that the computational power of GNNs is related to the unfolding trees, namely trees that can be constructed by visiting the graph from a given node. In this paper, we unify these two theories and prove that the Weisfeiler–Lehman test and the unfolding trees induce the same equivalence relationship on the graph nodes: such an equivalence exactly explains which nodes can or cannot be distinguished by a GNN. Moreover, it is proved that GNNs can approximate in probability, up to any precision, any function on graphs that respects the above mentioned equivalence relationship. These results provide a more comprehensive understanding of the computational power of GNNs in node classification/regression tasks. Index Terms—Graph Neural Networks, Weisfeiler–Lehman, unfolding trees, graph theory, deep learning, message passing, expressive power.