A Topological characterisation of Weisfeiler-Leman equivalence classes
This provides a theoretical foundation for understanding limitations in GNN expressiveness, which is crucial for researchers and practitioners in graph machine learning.
The paper characterizes the classes of graphs that Graph Neural Networks (GNNs) cannot distinguish using covering space theory, and generates the GraphCovers dataset containing arbitrarily many non-isomorphic graphs that GNNs fail to distinguish, with the number of indistinguishable graphs growing super-exponentially with node count.
Graph Neural Networks (GNNs) are learning models aimed at processing graphs and signals on graphs. The most popular and successful GNNs are based on message passing schemes. Such schemes inherently have limited expressive power when it comes to distinguishing two non-isomorphic graphs. In this article, we rely on the theory of covering spaces to fully characterize the classes of graphs that GNNs cannot distinguish. We then generate arbitrarily many non-isomorphic graphs that cannot be distinguished by GNNs, leading to the GraphCovers dataset. We also show that the number of indistinguishable graphs in our dataset grows super-exponentially with the number of nodes. Finally, we test the GraphCovers dataset on several GNN architectures, showing that none of them can distinguish any two graphs it contains.