# Online Bayesian Node Classification on Inductive Graphs under Distribution Shift

arXiv: https://arxiv.org/abs/2609.13655 · HTML: https://mlsanity.com/paper/2609.13655/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 23 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Variational Continual Learning](https://mlsanity.com/paper/1710.10628/) (2017)
- [Handling Distribution Shifts on Graphs: An Invariance Perspective](https://mlsanity.com/paper/2202.02466/) (2022)
- [Be Confident! Towards Trustworthy Graph Neural Networks via Confidence Calibration](https://mlsanity.com/paper/2109.14285/) (2021)
- [Bayesian graph convolutional neural networks for semi-supervised classification](https://mlsanity.com/paper/1811.11103/) (2018)
- [Bayesian Graph Neural Networks with Adaptive Connection Sampling](https://mlsanity.com/paper/2006.04064/) (2020)
- [Graph Neural Networks: Architectures, Stability and Transferability](https://mlsanity.com/paper/2008.01767/) (2020)
- [Variational Bayesian Last Layers](https://mlsanity.com/paper/2404.11599/) (2024)
- [Streaming Graph Neural Networks via Continual Learning](https://mlsanity.com/paper/2009.10951/) (2020)
- [Knowledge-Adaptation Priors](https://mlsanity.com/paper/2106.08769/) (2021)
- [Online scalable Gaussian processes with conformal prediction for guaranteed coverage](https://mlsanity.com/paper/2410.05444/) (2024)
- [Conformalized Gaussian processes for online uncertainty quantification over graphs](https://mlsanity.com/paper/2510.06181/) (2025)

## Abstract (arXiv)

On evolving graphs, node classifiers must satisfy two key requirements: inductive generalization to newly arriving nodes under distribution shift and calibrated uncertainty for safety-sensitive applications. Standard graph neural networks (GNNs) are typically trained once and address neither requirement. We adapt the Bayesian last-layer (BLL) model by placing random last-layer parameters on top of a deterministic GNN encoder for uncertainty quantification. The categorical softmax likelihood required for classification breaks Gaussian conjugacy, so neither the training posterior nor the test-time streaming update has a closed-form solution. To address both challenges, we introduce a variational Bayesian last-layer (VBLL) objective that jointly trains the encoder and an approximate last-layer posterior by maximizing an evidence lower bound with a Monte Carlo expected log-likelihood. At test time, we freeze the encoder and apply an online Laplace update to the last-layer posterior. This update corresponds to a power-prior Bayesian model with exponential forgetting and a Kullback-Leibler anchor to the training posterior. Across five node-classification benchmarks under distribution shift, online GVBLL is the only method to achieve the best accuracy and negative log-likelihood on every dataset. It improves accuracy by up to 17 percentage points on Cora and 14 percentage points on ogbn-arxiv over the strongest non-GVBLL baseline, while remaining competitive in calibration with MC Dropout, Deep Ensembles, Temperature Scaling, and Gaussian-process classifiers.