Graph Classification via Network Usable Information: From Representation Evaluation to Structure Selection

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.

Graph theory is inherently descriptive, capturing what relationships exist but not why they arise, because it treats edges as primitive constructs. This paper proposes a new explanatory framework for graph learning, where relationships emerge from latent continuous information entropy fields, and a graph becomes a discrete instantiation of an underlying field. To formalize this field, we introduce the Field-informed Graph Network (FGN). It learns a scalar field from node features and leverages it to modulate message passing. The information-theoretic objective balances structural fidelity with field smoothness, forming a self-reinforcing loop. In this loop, the field modulates information diffusion through field-modulated weighting, and the updated node representations iteratively refine the field. As a result, FGN learns by simulating its own co-evolution. Extensive experiments on node classification and graph classification benchmarks demonstrate superior performance, robustness to perturbations, and structurally coherent field representations.

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.