A fast topological approach for predicting anomalies in time-varying graphs

TL;DR

This paper introduces a fast, topological data analysis-based method for detecting anomalies in time-varying graphs, improving change point detection and price prediction without embedding graphs into metric spaces.

Contribution

It presents a novel, efficient framework using persistence diagrams and Betti functions for shape analysis of graphs, avoiding metric embedding and enhancing anomaly detection.

Findings

Improved change point detection rate in simulations

Up to 22% gain in cryptocurrency price anomaly prediction

Framework is stable against input noise

Abstract

Large time-varying graphs are increasingly common in financial, social and biological settings. Feature extraction that efficiently encodes the complex structure of sparse, multi-layered, dynamic graphs presents computational and methodological challenges. In the past decade, a persistence diagram (PD) from topological data analysis (TDA) has become a popular descriptor of shape of data with a well-defined distance between points. However, applications of TDA to graphs, where there is no intrinsic concept of distance between the nodes, remain largely unexplored. This paper addresses this gap in the literature by introducing a computationally efficient framework to extract shape information from graph data. Our framework has two main steps: first, we compute a PD using the so-called lower-star filtration which utilizes quantitative node attributes, and then vectorize it by averaging the associated Betti function over successive scale values on a one-dimensional grid. Our approach avoids embedding a graph into a metric space and has stability properties against input noise. In simulation studies, we show that the proposed vector summary leads to improved change point detection rate in time-varying graphs. In a real data application, our approach provides up to 22% gain in anomalous price prediction for the Ethereum cryptocurrency transaction networks.