Graph Neural Networks Are Not Continuous Across Graph Resolutions

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.

Graph neural networks (GNNs) are powerful tools used to analyze complex systems like molecules or social networks, but they suffer from a hidden blind spot. We reveal that GNNs fail to recognize when two graphs represent the exact same object at different levels of detail, causing them to generate completely different predictions. We trace this glitch to a fundamental limitation in how standard GNNs pass information between neighboring points. To solve this problem, we develop a mathematical modification that keeps the graph neural networks understanding stable and consistent across all resolution levels. We systematically validate this upgrade, proving it allows GNNs to reliably share knowledge between simple and highly detailed models to make AI predictions far more robust.