$\mathscr{N}$-WL: A New Hierarchy of Expressivity for Graph Neural Networks

The expressive power of Graph Neural Networks (GNNs) is fundamental for understanding their capabilities and limitations, i.e., what graph properties can or cannot be learnt by a GNN. Since standard GNNs have been characterised to be upper-bounded by the Weisfeiler-Lehman (1-WL) algorithm, recent attempts concentrated on developing more expressive GNNs in terms of the $k$-WL hierarchy, a well-established framework for graph isormorphism tests. In this work we show that, contrary to the widely accepted view, the $k$-WL hierarchy is not well-suited for measuring expressive GNNs. This is due to limitations that are inherent to high-dimensional WL algorithms such as the lack of a natural interpretation and high computational costs, which makes it difficult to draw any firm conclusions about the expressive power of GNNs beyond 1-WL. Thus, we propose a novel hierarchy of graph isomorphism tests, namely Neighbourhood WL ($\mathscr{N}$-WL), and also establish a new theorem on the equivalence of expressivity between induced connected subgraphs and induced subgraphs within this hierarchy. Further, we design a GNN model upon $\mathscr{N}$-WL, Graph Neighbourhood Neural Network (G3N), and empirically verify its expressive power on synthetic and real-world benchmarks.