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× speedups over full re-solving and roughly 106× speedups over gradient retraining. Structural-inversion experiments further quantify the privacy floor of exact retraining and the additional leakage of approximate graph-unlearning methods.
In the transductive setting, where the full graph is observed but node labels are only partially available, progress in semi-supervised node classification has largely focused on architectural innovation. In this paper, we revisit an orthogonal axis: the training objective. We start from a simple observation: transductive models produce predictions for every node during training, including nodes without labels. These unlabeled-node predictions may contain useful training signal, but standard supervised objectives discard them because no ground-truth labels are available. Inspired by the decomposition of cross-entropy into a label-dependent alignment term and a label-independent entropy term, we propose prediction confidence as a natural way to extract this signal in the absence of labels. This motivates Transductive Sharpening (TS): a loss-level modification that minimizes prediction entropy on unlabeled nodes while counterbalancing this effect on labeled nodes. We evaluate Transductive Sharpening across a wide range of node-classification benchmarks and observe consistent performance improvements without requiring any changes to the backbone architecture. Code is available at https://github.com/transductive-sharpening/tunedGNN.
Brown Zaz, Mar Gonzàlez I Català, Ferran Hernandez Caralt +2
We propose NetinfoGC, a framework for graph classification that extends the Network Usable Information (NUI) paradigm to graph-level learning. Unlike conventional graph neural network approaches that rely on end-to-end training of black-box embeddings, NetinfoGC constructs a family of permutation-invariant graph representations derived from propagation-based mechanisms and classical structural descriptors, including graph centrality measures. To evaluate representation quality, we introduce a training-free NUI estimation procedure based on clustering consistency with ground-truth labels, providing a proxy for task-relevant information without supervised learning. We further exploit the same representations using sparse-group LASSO regularization, enabling automatic selection of informative structural descriptors while suppressing redundant ones. Experiments on benchmark datasets show that classical centrality measures are highly competitive with learned propagation-based representations, and in several cases yield superior performance. Moreover, we observe a strong correlation between estimated NUI and downstream classification accuracy, validating NUI as an effective measure of representation utility. Overall, NetinfoGC provides a unified and interpretable framework for evaluating and exploiting graph representations without requiring end-to-end neural training.
Node representation learning has advanced rapidly, yet most existing methods rely on per-dataset training and hyperparameter tuning. This dataset-specific optimization comes from the difficulty of designing reusable graph models that generalize across diverse graph datasets. In this work, we introduce Node4All, a node representation learner applicable to arbitrary graph datasets without any dataset-specific optimization. Node4All is built on two complementary ideas. At the architectural level, we introduce the Channel Graph Transformer (CGT), which enables a single fixed parameterization to process arbitrary graph datasets. At the learning level, we propose a self-supervised learning based on a series of synthetic graphs. Together, these components enable generalization beyond individual datasets, which is infeasible with existing architectures and learning frameworks. We extensively evaluate Node4All on node classification across 25 benchmarks against 21 baselines, covering both supervised and self-supervised methods. Despite all baselines being trained and optimized for each dataset, a single Node4All, applied uniformly across the datasets, achieves a competitive ranking of 5th among 21 baselines. Moreover, Node4All supports one-shot and in-context learning with an appropriate predictor and outperforms recent graph foundation models (GFMs) in these settings. These results demonstrate that Node4All not only achieves reusability across arbitrary graph datasets, but also remains an effective solution in practice. Code and model checkpoints are available in https://github.com/dooho00/node4all.