📝 SummarXiv

Abstract

The success of graph neural networks (GNNs) has motivated theoretical studies on their expressive power, often through alignments with the Weisfeiler-Lehman (WL) tests. However, such analyses typically focus on the ability of GNNs to distinguish between graph structures, rather than to compute or approximate specific function classes. The latter is more commonly studied in machine learning theory, including results such as the Turing completeness of recurrent networks and the universal approximation property of feedforward networks. We argue that using well-defined computational models, such as a modified CONGEST model with clearly specified preprocessing and postprocessing, offers a more sound framework for analyzing GNN expressiveness. Within this framework, we show that allowing unrestricted preprocessing or incorporating externally computed features, while claiming that these precomputations enhance the expressiveness, can sometimes lead to problems. We also show that the lower bound on a GNN's capacity (depth multiplied by width) to simulate one iteration of the WL test actually grows nearly linearly with graph size, indicating that the WL test is not locally computable and is misaligned with message-passing GNNs. Despite these negative results, we also present positive results that characterize the effects of virtual nodes and edges from a computational model perspective. Finally, we highlight several open problems regarding GNN expressiveness for further exploration.