Closed-Form Node Classification with Exact Graph Unlearning

2026-05-25

AI Summary

This work addresses the inefficiency of conventional graph neural networks (GNNs), which rely on time-consuming gradient-based training and struggle to achieve exact graph unlearning. The authors propose the first deterministic, closed-form framework for node classification that enables precise unlearning at the level of labels, features, edges, nodes, and subgraphs, with guaranteed K-hop locality. Leveraging graph homophily, the method adaptively selects between two strategies: for homophilic graphs, it employs simplified graph convolution (SGC) propagation combined with ridge regression; for heterophilic graphs, it introduces LCF-Net, which integrates layer-wise closed-form feature refinement with a Gaussian kernel ridge regression head. Evaluated across 14 benchmarks, the approach outperforms standard 2-layer GCN, GraphSAGE, and GAT on all 9 datasets tested and matches or exceeds tuned deep models on 9 of 12 small-scale datasets. On ogbn-arxiv, local updates are 21–45× faster than full retraining and approximately 10⁶× faster than gradient-based retraining.

Abstract

Graph neural networks for node classification are typically trained by gradient descent over hundreds or thousands of epochs. Recent work has shown that, when properly tuned, classic GCN/SAGE/GAT architectures can match graph transformers on many node-classification benchmarks. We ask a complementary question: how much of this performance can be recovered by deterministic closed-form solvers, and what guarantees does this enable? We introduce a routed closed-form framework selected by adjusted homophily. For assortative graphs, we use SGC-style propagation followed by Ridge regression; for heterophilous graphs, we introduce LCF-Net, a layer-wise closed-form graph feature-refinement network whose per-layer Ridge solves are capped by a Gaussian kernel-Ridge head. Across 14 benchmarks, including ogbn-arxiv and ogbn-proteins, our closed-form predictors match or beat the best vanilla 2-layer GCN/SAGE/GAT on 9 of 9 measured datasets, tie tuned deep recipes within one standard deviation on 9 of 12 small benchmarks, and exceed the OGB-leaderboard plain GCN on both large graphs. The remaining heterophilous gap closely tracks the gain from vanilla 2-layer to deep SAGE, suggesting that the residual difference is primarily architectural. Because our predictors are explicit solutions of deterministic linear systems, modified graph inputs can be re-solved to obtain retrain-equivalent parameters. We formalize exact graph-object unlearning for label, feature, edge, node, and subgraph modifications, prove K-hop locality for Ridge components, and verify exactness across 109 configurations. On ogbn-arxiv, localized updates give $21$--$45\times$ speedups over full re-solving and roughly $10^{6}\times$ speedups over gradient retraining. Structural-inversion experiments further quantify the privacy floor of exact retraining and the additional leakage of approximate graph-unlearning methods.

Problem

• node classification
• closed-form solver
• graph unlearning
• exact retraining
• graph neural networks

Innovation

• closed-form learning
• graph unlearning
• exact retraining
• homophily-aware routing
• LCF-Net