Topological Analysis of Temporal Hypergraphs

TL;DR

This paper introduces a topological data analysis approach using zigzag persistence to study the evolving structure of temporal hypergraphs, providing deeper insights into their dynamics beyond traditional snapshot methods.

Contribution

It applies zigzag persistence from TDA to analyze the topological evolution of temporal hypergraphs, capturing persistent features over time.

Findings

Topological features reveal system dynamics in cyber security and social networks.

The method uncovers persistent components influencing the underlying systems.

Temporal hypergraph topology changes correlate with system behavior.

Abstract

In this work we study the topological properties of temporal hypergraphs. Hypergraphs provide a higher dimensional generalization of a graph that is capable of capturing multi-way connections. As such, they have become an integral part of network science. A common use of hypergraphs is to model events as hyperedges in which the event can involve many elements as nodes. This provides a more complete picture of the event, which is not limited by the standard dyadic connections of a graph. However, a common attribution to events is temporal information as an interval for when the event occurred. Consequently, a temporal hypergraph is born, which accurately captures both the temporal information of events and their multi-way connections. Common tools for studying these temporal hypergraphs typically capture changes in the underlying dynamics with summary statistics of snapshots sampled in a sliding window procedure. However, these tools do not characterize the evolution of hypergraph structure over time, nor do they provide insight on persistent components which are influential to the underlying system. To alleviate this need, we leverage zigzag persistence from the field of Topological Data Analysis (TDA) to study the change in topological structure of time-evolving hypergraphs. We apply our pipeline to both a cyber security and social network dataset and show how the topological structure of their temporal hypergraphs change and can be used to understand the underlying dynamics.