Efficient Dynamic Algorithms for Graph Neural Networks with Non-Linear Propagation

2026-09-26

AI Summary

This study addresses the inefficiency of maintaining nonlinear GNN propagation representations in dynamic graph settings, where edge insertions and deletions typically necessitate full recomputation. To overcome this limitation, the work proposes a residual-based dynamic update algorithm that efficiently maintains node representations through local error pushing. Technically, it integrates potential analysis, degree-scaled norms, and low-rank matrix inverse updates to transcend linear constraints, achieving an amortized O(1/ε) update complexity for nonlinear propagation without requiring stochasticity assumptions, while also providing exact algorithms for the linear case. Experiments on benchmark datasets demonstrate that the proposed approach simultaneously attains high approximation accuracy and efficient dynamic updating capabilities, offering both rigorous theoretical guarantees and a practical solution for nonlinear dynamic graph learning.

Abstract

Graph Neural Networks (GNNs) are widely used for representation learning on graphs, but most methods assume static topologies, making them inefficient on evolving networks where edges change over time. Existing dynamic approaches either model graph evolution through temporal GNN architectures without focusing on efficient dynamic maintenance, or are restricted to linear propagation models based on Personalized PageRank. In this work, we study how to efficiently maintain node representations for non-linear GNN propagation under edge insertions and deletions. The propagation has no learned parameters, and only a classifier applied afterward is trained. For a broad class of standard activation functions, we develop a residual-based dynamic algorithm that selectively propagates local errors via push operations, maintaining an approximation to the evolving fixed point without full recomputation. We prove that our method achieves amortized $O(1/\epsilon)$ update time per graph change under a degree-normalized error guarantee. Our approach uses a potential-based analysis in a degree-scaled norm and, in contrast to prior work on the linear case, requires no randomness assumptions on either the update sequence or the input vector. For the linear special case, we additionally provide an exact dynamic algorithm via low-rank matrix inverse updates. Experiments on benchmark datasets show that incorporating non-linearity improves accuracy while preserving efficient update performance, yielding a scalable and theoretically grounded method for maintaining this propagation on dynamic graphs.

Problem

• Graph Neural Networks
• Dynamic Graphs
• Non-Linear Propagation
• Node Representations
• Efficient Maintenance

Innovation

• Dynamic Graph Neural Networks
• Non-linear Propagation
• Residual-based Algorithm
• Amortized Update Time
• Potential-based Analysis