How Powerful are K-hop Message Passing Graph Neural Networks

Jiarui Feng, Yixin Chen, Fuhai Li, Anindya Sarkar, Muhan Zhang

Keywords: Graph Neural Network, Expressive Power, Graph Isomorphism, Weisfeiler-Lehman

TL;DR: In this paper, we first theoretically analyze the expressive power and the limitation of K-hop message passing graph neural networks. Then, we propose a novel method to improve the K-hop message passing framework.

Abstract: The most popular design paradigm for Graph Neural Networks (GNNs) is 1-hop message passing---aggregating information from 1-hop neighbors repeatedly. However, the expressive power of 1-hop message passing is bounded by the Weisfeiler-Lehman (1-WL) test. Recently, researchers extended 1-hop message passing to $K$-hop message passing by aggregating information from $K$-hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive power of $K$-hop message passing. In this work, we theoretically characterize the expressive power of $K$-hop message passing. Specifically, we first formally differentiate two different kernels of $K$-hop message passing which are often misused in previous works. We then characterize the expressive power of $K$-hop message passing by showing that it is more powerful than 1-WL and can distinguish almost all regular graphs. Despite the higher expressive power, we show that $K$-hop message passing still cannot distinguish some simple regular graphs and its expressive power is bounded by 3-WL. To further enhance its expressive power, we introduce a KP-GNN framework, which improves $K$-hop message passing by leveraging the peripheral subgraph information in each hop. We show that KP-GNN can distinguish many distance regular graphs which could not be distinguished by previous distance encoding or 3-WL methods. Experimental results verify the expressive power and effectiveness of KP-GNN. KP-GNN achieves competitive results across all benchmark datasets.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2205.13328/code)