# Graph Anomaly Detection as Finite-Horizon Control: Training-Free Scoring via Empirical Bayes

arXiv: https://arxiv.org/abs/2609.38424 · HTML: https://mlsanity.com/paper/2609.38424/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 49 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Enhancing Graph Neural Network-based Fraud Detectors against Camouflaged Fraudsters](https://mlsanity.com/paper/2008.08692/) (2020)
- [A Comprehensive Survey on Graph Anomaly Detection with Deep Learning](https://mlsanity.com/paper/2106.07178/) (2021)
- [Rethinking Graph Neural Networks for Anomaly Detection](https://mlsanity.com/paper/2205.15508/) (2022)
- [Graph-based Anomaly Detection and Description: A Survey](https://mlsanity.com/paper/1404.4679/) (2014)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Anomaly Detection on Attributed Networks via Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2103.00113/) (2021)
- [Deep Isolation Forest for Anomaly Detection](https://mlsanity.com/paper/2206.06602/) (2022)
- [AnomalyDAE: Dual autoencoder for anomaly detection on attributed networks](https://mlsanity.com/paper/2002.03665/) (2020)
- [Alleviating the Inconsistency Problem of Applying Graph Neural Network to Fraud Detection](https://mlsanity.com/paper/2005.00625/) (2020)
- [Generative and Contrastive Self-Supervised Learning for Graph Anomaly Detection](https://mlsanity.com/paper/2108.09896/) (2021)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [DGraph: A Large-Scale Financial Dataset for Graph Anomaly Detection](https://mlsanity.com/paper/2207.03579/) (2022)
- [Truncated Affinity Maximization: One-class Homophily Modeling for Graph Anomaly Detection](https://mlsanity.com/paper/2306.00006/) (2023)
- [Image Restoration Through Generalized Ornstein-Uhlenbeck Bridge](https://mlsanity.com/paper/2312.10299/) (2023)
- [SpecAE: Spectral AutoEncoder for Anomaly Detection in Attributed Networks](https://mlsanity.com/paper/1908.03849/) (2019)
- [PyGOD: A Python Library for Graph Outlier Detection](https://mlsanity.com/paper/2204.12095/) (2022)
- [Semi-supervised Credit Card Fraud Detection via Attribute-Driven Graph Representation](https://mlsanity.com/paper/2412.18287/) (2024)
- [Label Information Enhanced Fraud Detection against Low Homophily in Graphs](https://mlsanity.com/paper/2302.10407/) (2023)
- [Cauchy combination test: a powerful test with analytic p-value calculation under arbitrary dependency structures](https://mlsanity.com/paper/1808.09011/) (2018)
- [Revisiting Graph-Based Fraud Detection in Sight of Heterophily and Spectrum](https://mlsanity.com/paper/2312.06441/) (2023)
- [UniDB: A Unified Diffusion Bridge Framework via Stochastic Optimal Control](https://mlsanity.com/paper/2502.05749/) (2025)
- [Demystifying Fraudulent Transactions and Illicit Nodes in the Bitcoin Network for Financial Forensics](https://mlsanity.com/paper/2306.06108/) (2023)
- [PyOD 2: A Python Library for Outlier Detection with LLM-powered Model Selection](https://mlsanity.com/paper/2412.12154/) (2024)
- [DiffGAD: A Diffusion-based Unsupervised Graph Anomaly Detector](https://mlsanity.com/paper/2410.06549/) (2024)
- [Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations](https://mlsanity.com/paper/2506.06907/) (2025)

## Abstract (arXiv)

Node-level graph anomaly detection (GAD) identifies nodes whose attributes and interactions deviate from dominant graph regularities. Existing GAD models encode normality and anomaly scoring indirectly through architectures, message passing, reconstruction or contrastive objectives, and tuned score families. This entangles graph trust (how strongly graph structure should define normality), graph-spectral weighting, and anomaly-score choice, yielding scores that are costly, opaque, and unstable across graph regimes. We propose EB-GAD (Empirical-Bayes GAD), a training-free framework that models normality as graph-aware generalized Ornstein-Uhlenbeck (GOU) relaxation toward a graph-filtered template. Empirical Bayes fits the graph precision from the residual-field likelihood; the GOU then turns scoring into a closed-form finite-horizon control energy, the minimum effort to steer a feature-neutral node to its observed endpoint along graph-spectral relaxation. Sweeping relaxation horizon and endpoint tolerance yields a bank of scores that share one fitted prior: equilibrium Mahalanobis scoring is one limit, while finite-horizon control-energy and scale-normalized ratio scores reveal anomalies that static equilibrium scoring can mask. A label-free selector chooses the score family from feature homophily, edge density, and feature dimension, then ranks candidates by fitted-null deviation and rank stability. On 11 benchmarks and without labels at any step, EB-GAD has the best or tied-best AUROC on 9: the four financial fraud networks (up to 3.7M nodes), the YelpChi and Amazon review graphs, Weibo, Reddit and Facebook, with margins of up to 21.7 points. It is second on BlogCatalog and ACM.