Graph Neural Networks Are Not Continuous Across Graph Resolutions
This work addresses a fundamental limitation in GNNs' ability to consistently process graphs at different resolutions, which is crucial for applications requiring robust generalization and integration of multi-resolution data.
This paper demonstrates that graph neural networks (GNNs) are not continuous across different graph resolutions, leading to significantly different latent representations for graphs representing the same object at varying scales. The authors identify a structural obstruction in information-propagation schemes as the cause and propose a modification to standard GNN architectures to achieve continuity across scales.
We show that contrary to conventional wisdom in the community, graph neural networks (GNNs) are not continuous with respect to all natural modes of graph convergence. As a result, GNNs may generate substantially different latent representations for graphs that are very similar. In particular they assign vastly different latent embeddings to graphs that represent the same underlying object at different resolution scales. We trace this failure of continuity back to a structural obstruction arising from commonly used information-propagation schemes. Building on this insight we then derive a principled modification to standard GNN architectures which equips models with continuity across scales. The proposed modification enables consistent integration of distinct resolutions and reliable generalization between them. We systematically validate our theoretical findings in a wide range of numerical experiments.