# A Unified Uncertainty Representation for Graph Neural Networks via Doubly-Spectral Stochastic Expansion

arXiv: https://arxiv.org/abs/2609.35703 · HTML: https://mlsanity.com/paper/2609.35703/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 38 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Beyond Low-frequency Information in Graph Convolutional Networks](https://mlsanity.com/paper/2101.00797/) (2021)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [Enhancing Graph Neural Network-based Fraud Detectors against Camouflaged Fraudsters](https://mlsanity.com/paper/2008.08692/) (2020)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [GOOD: A Graph Out-of-Distribution Benchmark](https://mlsanity.com/paper/2206.08452/) (2022)
- [Graph Posterior Network: Bayesian Predictive Uncertainty for Node Classification](https://mlsanity.com/paper/2110.14012/) (2021)
- [Energy-based Out-of-Distribution Detection for Graph Neural Networks](https://mlsanity.com/paper/2302.02914/) (2023)
- [Energy-based Epistemic Uncertainty for Graph Neural Networks](https://mlsanity.com/paper/2406.04043/) (2024)
- [Accurate and Scalable Estimation of Epistemic Uncertainty for Graph Neural Networks](https://mlsanity.com/paper/2401.03350/) (2024)
- [The Deep Arbitrary Polynomial Chaos Neural Network or how Deep Artificial Neural Networks could benefit from Data-Driven Homogeneous Chaos Theory](https://mlsanity.com/paper/2306.14753/) (2023)
- [Topology-Aware Dynamic Reweighting for Distribution Shifts on Graph](https://mlsanity.com/paper/2406.01066/) (2024)

## Abstract (arXiv)

Reliable deployment of graph neural networks requires calibration, out-of-distribution (OOD) detection, and robustness to distribution shift, yet existing methods address these needs with separate models and objectives. We model uncertain node embeddings as random graph signals: graph Fourier filters capture structural variation, and a scalar orthogonal-polynomial chaos coordinate captures latent stochastic variation. The resulting doubly-spectral stochastic (DSS) expansion supplies task-matched readouts from one representation: the mean coefficient encodes class evidence for the energy-based OOD score, the higher-order coefficients encode structured logit variation, and quadrature averaging over the chaos coordinate defines the single predictive distribution used for prediction and calibration. A capacity theorem shows that, under a full-rank feature assumption, a restricted subfamily matches the chaos coefficients of any Gaussian-latent random graph signal, with exponentially decaying truncation error under a growth condition; the task-level claims are established empirically. DSS-GNN has two deployment modes: standalone, or as a residual branch beside a deterministic encoder (DSS-Hybrid). Standalone DSS-GNN achieves the lowest Brier score among the compared uncertainty-aware baselines on all 14 node classification benchmarks without post-hoc correction; DSS-Hybrid achieves the best AUROC on most node-OOD settings, competitive cross-graph OOD detection, and the strongest shifted accuracy on all 7 GOOD concept-shift benchmarks under standard empirical risk minimization (ERM). Cross-evaluating both modes on all three tasks shows that each remains effective on the other's tasks, with documented exceptions, and yields explicit deployment guidance.