Closed-Form Node Classification with Exact Graph Unlearning

Graph neural networks for node classification are typically trained by gradient descent over hundreds or thousands of epochs. Recent work shows that carefully tuned classic GCN, GraphSAGE and GAT architectures can rival graph transformers. We ask a complementary question: can closed-form solvers recover this predictive strength and make exact graph unlearning practical? One validation-selected protocol combines multiscale diffusion, nonlinear graph kernels and training-label context with analytical readouts. Across generic and task-aware tracks, **closed-form predictors exceed or come within one point** of the strongest reproduced tuned GNN on all 16 reported benchmarks. The generic protocol achieves this on 14/16, including both OGB graphs; task-aware constructions surpass the remaining two references, setting a new Minesweeper state of the art at 99.95 ROC-AUC with an 11 ms clue-based CPU fit, and reaching 91.60 accuracy on Roman-empire. The generic protocol also exceeds cited retuned GraphGPS and SGFormer means on 14/15 and 12/15 comparisons. We derive exact graph-unlearning procedures that recover the result of retraining the same model from scratch after deleting labels, features, edges, nodes or subgraphs. They reuse unaffected information and recompute only what changes. Across 280 deterministic verification cases, 232 match retrained prediction scores exactly; 43 double-precision label updates differ by less than $2.6\times 10^{-11}$, and five Questions updates preserve every predicted class against the single-precision reference. Timed in default GPU mode, cached one-label model updates are a **median 93× faster** than tuned-GNN training steps alone across fifteen same-hardware comparisons, with speedups up to approximately 20,000×. Closed-form fitting therefore supports both strong node prediction and exact, verifiable unlearning.