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Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection

Souhail Abdelmouaiz Sadat, Mohamed Yacine Touahria Miliani, Khadidja Hab El Hames, Hamida Seba, Mohammed Haddad

TL;DR

This work assesses the suitability of hyperbolic graph embeddings for anomaly detection, arguing that negative curvature better captures hierarchical and complex graph structures than Euclidean space. It surveys traditional and deep-learning-based hyperbolic embedding methods, and introduces a practical taxonomy and an open-source library for benchmarking. A unified anomaly-detection framework pairs hyperbolic embeddings with conventional detectors, enabling reproducible evaluation across diverse datasets. Empirically, hyperbolic methods generally outperform Euclidean baselines, highlighting the practical potential of hyperbolic geometry for robust anomaly detection in real-world networks.

Abstract

This survey reviews hyperbolic graph embedding models, and evaluate them on anomaly detection, highlighting their advantages over Euclidean methods in capturing complex structures. Evaluating models like \textit{HGCAE}, \textit{\(\mathcal{P}\)-VAE}, and \textit{HGCN} demonstrates high performance, with \textit{\(\mathcal{P}\)-VAE} achieving an F1-score of 94\% on the \textit{Elliptic} dataset and \textit{HGCAE} scoring 80\% on \textit{Cora}. In contrast, Euclidean methods like \textit{DOMINANT} and \textit{GraphSage} struggle with complex data. The study emphasizes the potential of hyperbolic spaces for improving anomaly detection, and provides an open-source library to foster further research in this field.

Hyperbolic Graph Embeddings: a Survey and an Evaluation on Anomaly Detection

TL;DR

This work assesses the suitability of hyperbolic graph embeddings for anomaly detection, arguing that negative curvature better captures hierarchical and complex graph structures than Euclidean space. It surveys traditional and deep-learning-based hyperbolic embedding methods, and introduces a practical taxonomy and an open-source library for benchmarking. A unified anomaly-detection framework pairs hyperbolic embeddings with conventional detectors, enabling reproducible evaluation across diverse datasets. Empirically, hyperbolic methods generally outperform Euclidean baselines, highlighting the practical potential of hyperbolic geometry for robust anomaly detection in real-world networks.

Abstract

This survey reviews hyperbolic graph embedding models, and evaluate them on anomaly detection, highlighting their advantages over Euclidean methods in capturing complex structures. Evaluating models like \textit{HGCAE}, \textit{-VAE}, and \textit{HGCN} demonstrates high performance, with \textit{-VAE} achieving an F1-score of 94\% on the \textit{Elliptic} dataset and \textit{HGCAE} scoring 80\% on \textit{Cora}. In contrast, Euclidean methods like \textit{DOMINANT} and \textit{GraphSage} struggle with complex data. The study emphasizes the potential of hyperbolic spaces for improving anomaly detection, and provides an open-source library to foster further research in this field.

Paper Structure

This paper contains 22 sections, 9 equations, 2 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Graph Embedding in the Euclidean Space
  3. 1- Matrix Factorization-Based Methods:
  4. 2- Random Walk-Based Methods:
  5. 3- Deep Learning-Based Methods:
  6. Hyperbolic Graph Embedding
  7. Differential Geometry
  8. Hyperbolic Geometry
  9. Lorentz Model
  10. Poincaré ball model
  11. Klein model
  12. Taxonomy of Hyperbolic Graph Embedding Methods
  13. Direct optimization-based methods:
  14. Tree Approximation-Based Methods:
  15. Multidimensional Scaling-Based Methods:
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Taxonomy of hyperbolic graph embedding methods.
  • Figure 2: Global architecture of the anomaly detection framework.

Theorems & Definitions (10)

  • Definition 1: Topological Space
  • Definition 2: Manifold
  • Definition 3: Smooth Manifold
  • Definition 4: Tangent Space
  • Definition 5: Riemannian Manifold
  • Definition 6: Geodesics
  • Definition 7: Exponential Map
  • Definition 8: Logarithmic Map
  • Definition 9: Gaussian Curvature
  • Definition 10: Sectional curvature