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

Qing Wang, Dillon Ze Chen, Asiri Wijesinghe, Shouheng Li, Muhammad Farhan

Keywords: Graph neural network, Weisfeiler-Lehman algorithm, k-WL hierarchy, graph classification

Abstract: 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.

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