Graph Neural Networks Are Not Continuous Across Graph Resolutions

2026-05-29

AI Summary

This work addresses the lack of continuity in graph neural networks (GNNs) under varying graph resolutions, which leads to inconsistent embeddings for the same object across different scales. The study is the first to identify that this issue stems from a structural flaw in existing message-passing mechanisms when applied under natural graph convergence patterns. To resolve this, the authors propose a novel GNN architecture grounded in graph convergence theory that ensures cross-scale continuity. Extensive experiments demonstrate that the proposed method effectively achieves consistent representation fusion and reliable generalization across multi-resolution graphs, significantly enhancing model robustness and transferability in scenarios involving scale variation.

Abstract

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.

Problem

• Graph Neural Networks
• Continuity
• Graph Resolution
• Latent Representations
• Graph Convergence

Innovation

• graph neural networks
• continuity
• graph resolution
• scale invariance
• information propagation