Rethinking the Expressiveness of GNNs: A Computational Model Perspective

Graph Neural Networks (GNNs) are extensively employed in graph machine learning, with considerable research focusing on their expressiveness. Current studies often assess GNN expressiveness by comparing them to the Weisfeiler-Lehman (WL) tests or classical graph algorithms. However, we identify three key issues in existing analyses: (1) some studies use preprocessing to enhance expressiveness but overlook its computational costs; (2) some claim the anonymous WL test's limited power while enhancing expressiveness using non-anonymous features, creating a mismatch; and (3) some characterize message-passing GNNs (MPGNNs) with the CONGEST model but make unrealistic assumptions about computational resources, allowing $\textsf{NP-Complete}$ problems to be solved in $O(m)$ depth. We contend that a well-defined computational model is urgently needed to serve as the foundation for discussions on GNN expressiveness. To address these issues, we introduce the Resource-Limited CONGEST (RL-CONGEST) model, incorporating optional preprocessing and postprocessing to form a framework for analyzing GNN expressiveness. Our framework sheds light on computational aspects, including the computational hardness of hash functions in the WL test and the role of virtual nodes in reducing network capacity. Additionally, we suggest that high-order GNNs correspond to first-order model-checking problems, offering new insights into their expressiveness.

GSN: A Universal Graph Neural Network Inspired by Spring Network

The design of universal Graph Neural Networks (GNNs) that operate on both homophilous and heterophilous graphs has received increased research attention in recent years. Existing heterophilous GNNs, particularly those designed in the spatial domain, lack a convincing theoretical or physical motivation. In this paper, we propose the Graph Spring Network (GSN), a universal GNN model that works for both homophilous and heterophilous graphs, inspired by spring networks and metric learning. We show that the GSN framework interprets many existing GNN models from the perspective of spring potential energy minimization with various metrics, which gives these models strong physical motivations. We also conduct extensive experiments to demonstrate our GSN framework's superior performance on real-world homophilous and heterophilous data sets.