HLSAD: Hodge Laplacian-based Simplicial Anomaly Detection

2025-05-30

AI Summary

Traditional graph-based anomaly detection struggles to model higher-order interactions, limiting its ability to identify complex structural anomalies. This paper addresses this challenge by proposing the first higher-order anomaly detection framework for temporal simplicial complexes, grounded in Hodge Laplacian spectral analysis. It models time-varying high-dimensional interactions via simplicial complexes and leverages the spectral properties of the Hodge Laplacian operator to characterize higher-order topological changes, enabling precise localization of event-type anomalies and change points. The method overcomes the representational limitations of graph-based approaches while balancing expressive power and computational efficiency. Extensive experiments on multiple synthetic and real-world datasets demonstrate significant performance gains over state-of-the-art graph-based methods, empirically validating the critical role of higher-order topological structure in anomaly detection.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In this paper, we propose HLSAD, a novel method for detecting anomalies in time-evolving simplicial complexes. While traditional graph anomaly detection techniques have been extensively studied, they often fail to capture changes in higher-order interactions that are crucial for identifying complex structural anomalies. These higher-order interactions can arise either directly from the underlying data itself or through graph lifting techniques. Our approach leverages the spectral properties of Hodge Laplacians of simplicial complexes to effectively model multi-way interactions among data points. By incorporating higher-dimensional simplicial structures into our method, our method enhances both detection accuracy and computational efficiency. Through comprehensive experiments on both synthetic and real-world datasets, we demonstrate that our approach outperforms existing graph methods in detecting both events and change points.

Problem

• Detects anomalies in time-evolving simplicial complexes
• Captures higher-order interactions missed by graph methods
• Uses Hodge Laplacians to model multi-way interactions

Innovation

• Uses Hodge Laplacians for anomaly detection
• Models higher-order interactions effectively
• Enhances accuracy and computational efficiency