# Combining Euclidean and Hyperbolic Representations for Node-level Anomaly Detection

arXiv: https://arxiv.org/abs/2510.11827 · HTML: https://mlsanity.com/paper/2510.11827/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 47 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 30 of 30)

- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [GCC: Graph Contrastive Coding for Graph Neural Network Pre-Training](https://mlsanity.com/paper/2006.09963/) (2020)
- [Predicting Dynamic Embedding Trajectory in Temporal Interaction Networks](https://mlsanity.com/paper/1908.01207/) (2019)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [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)
- [Anomaly Detection on Attributed Networks via Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2103.00113/) (2021)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [AnomalyDAE: Dual autoencoder for anomaly detection on attributed networks](https://mlsanity.com/paper/2002.03665/) (2020)
- [BOND: Benchmarking Unsupervised Outlier Node Detection on Static Attributed Graphs](https://mlsanity.com/paper/2206.10071/) (2022)
- [GADBench: Revisiting and Benchmarking Supervised Graph Anomaly Detection](https://mlsanity.com/paper/2306.12251/) (2023)
- [Graph Anomaly Detection via Multi-Scale Contrastive Learning Networks with Augmented View](https://mlsanity.com/paper/2212.00535/) (2022)
- [GOOD-D: On Unsupervised Graph Out-Of-Distribution Detection](https://mlsanity.com/paper/2211.04208/) (2022)
- [GAD-NR: Graph Anomaly Detection via Neighborhood Reconstruction](https://mlsanity.com/paper/2306.01951/) (2023)
- [Truncated Affinity Maximization: One-class Homophily Modeling for Graph Anomaly Detection](https://mlsanity.com/paper/2306.00006/) (2023)
- [PyGOD: A Python Library for Graph Outlier Detection](https://mlsanity.com/paper/2204.12095/) (2022)
- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [PREM: A Simple Yet Effective Approach for Node-Level Graph Anomaly Detection](https://mlsanity.com/paper/2310.11676/) (2023)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [HGWaveNet: A Hyperbolic Graph Neural Network for Temporal Link Prediction](https://mlsanity.com/paper/2304.07302/) (2023)
- [Dual-Geometric Space Embedding Model for Two-View Knowledge Graphs](https://mlsanity.com/paper/2209.08767/) (2022)
- [GraphSAC: Detecting anomalies in large-scale graphs](https://mlsanity.com/paper/1910.09589/) (2019)
- [SmoothGNN: Smoothing-aware GNN for Unsupervised Node Anomaly Detection](https://mlsanity.com/paper/2405.17525/) (2024)
- [Reinforcement Neighborhood Selection for Unsupervised Graph Anomaly Detection](https://mlsanity.com/paper/2312.05526/) (2023)
- [Normality Learning-based Graph Anomaly Detection via Multi-Scale Contrastive Learning](https://mlsanity.com/paper/2309.06034/) (2023)
- [HC-GLAD: Dual Hyperbolic Contrastive Learning for Unsupervised Graph-Level Anomaly Detection](https://mlsanity.com/paper/2407.02057/) (2024)
- [Three Revisits to Node-Level Graph Anomaly Detection: Outliers, Message Passing and Hyperbolic Neural Networks](https://mlsanity.com/paper/2403.04010/) (2024)

## Abstract (arXiv)

Node-level anomaly detection (NAD) is challenging due to diverse structural patterns and feature distributions. As such, NAD is a critical task with several applications which range from fraud detection, cybersecurity, to recommendation systems. We introduce Janus, a framework that jointly leverages Euclidean and Hyperbolic Graph Neural Networks to capture complementary aspects of node representations. Each node is described by two views, composed by the original features and structural features derived from random walks and degrees, then embedded into Euclidean and Hyperbolic spaces. A multi Graph-Autoencoder framework, equipped with a contrastive learning objective as regularization term, aligns the embeddings across the Euclidean and Hyperbolic spaces, highlighting nodes whose views are difficult to reconcile and are thus likely anomalous. Experiments on four real-world datasets show that Janus consistently outperforms shallow and deep baselines, empirically demonstrating that combining multiple geometric representations provides a robust and effective approach for identifying subtle and complex anomalies in graphs.