Interpretable structural temperature scaling for calibrating graph neural networks and point-cloud segmentation

For graph node classification, calibrated class probabilities are needed when confidence scores, usually the maximum predicted class probability, are used to rank predictions, defer uncertain nodes to human review, or control risk. Existing post-hoc calibrators either apply one global temperature or use graph-aware modules without a principled structural form. We study how local graph structure should enter node-level calibration. Our first results show that a logit-only temperature rule is insufficient when nodes with identical logits but different local homophily require different optimal temperatures. We then analyze a population-concentration contextual stochastic block model with Gaussian features and a one-layer linear GCN. Under equidistant class means, the Bayes posterior over class-template scores is a temperature-scaled softmax whose inverse-temperature is governed by a homophily-dependent signal strength. In the positive-signal homophilic regime, the resulting temperature decreases approximately inversely with normalized local homophily. This law motivates Homophily-aware Temperature Scaling (HoTS), a simple post-hoc calibrator that assigns each node a positive scalar temperature from entropy-based logit concentration and estimated local homophily. HoTS has three temperature parameters, preserves the predicted class, and learns the strength of the structural correction from calibration data. Across 18 node-classification benchmarks, two GNN backbones, and eight calibration baselines, HoTS achieves the best mean Expected Calibration Error (ECE) of 4.79%, the best average rank, and the most reliable confidence ranking in selective classification. Code is available at https://github.com/inu0104/HoTS.

Graph neural networks (GNNs) are increasingly deployed in real-world applications where distribution shift is un-avoidable. However, how such shifts affect model calibration, defined as the agreement between predictive confidence and actual accuracy, remains poorly understood, and existing graph calibration methods typically rely on labeled validation data from the deployment distribution. In this work, I present the first closed-form theoretical characterization of GNN calibration under distribution shift. I show that calibration is governed by a single scalar quantity that explicitly depends on structural changes between the source and target graphs, as well as feature quality. This characterization precisely identifies when a model becomes over-confident, under-confident, or remains calibrated, and directly yields the optimal temperature scaling strategy. I further extend the analysis to graph convolutional networks with symmetric normalization, multi-class classification, and covariate shift, and derive a theoretical upper bound on the expected calibration error. My analysis also reveals that, under homogeneous distribution shift, a single global temperature is theoretically optimal, providing a principled explanation for why more complex node-wise recalibration methods offer no additional benefit. Building on these theoretical insights, I propose STAC, a source-free, label-free calibration method. Experiments on synthetic benchmarks demonstrate substantial calibration improvements, while evaluations on five real-world graph datasets show that reliable calibration without target labels remains challenging despite the strong predictive power of the theory.

Modern neural networks are often miscalibrated, with a tendency to overconfidence. Existing train-time calibration methods largely modify task losses or calibration penalties, leaving neighborhood structure in learned representations underexploited. We introduce graph smoothing as a general principle for train-time calibration, which encourages similar predictive distributions across neighboring samples in representation space. We analyze the effects of graph smoothing, deriving bounds that connect predictive divergence between neighboring samples to local confidence variation and to the propagation of pointwise calibration error, and characterize the conditions under which smoothing can or cannot improve calibration. In light of this analysis, we propose \modelNoSpace, a graph-based train-time regularizer that penalizes the Jensen--Shannon divergence between predictive distributions of neighboring samples. We present a thorough empirical analysis, showing that across standard calibration benchmarks, \model improves predictive quality, and the improvement is complementary to post-hoc calibration: after temperature scaling, \model attains the lowest NLL of all evaluated train-time methods in seven of the eight image and tabular settings. These findings demonstrate the value of graph smoothing over learned representations for neural network calibration.