Topological Analysis of Temporal Hypergraphs
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 in comparison to the standard dyadic connection limitation 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 as well as their multi-way connections. Common tools for studying these temporal hypergraphs typically use summary statistics of snapshots from a sliding window procedure to capture changes in the underlying dynamics. However, these do not provide insight into how the changing structure of the hypergraph evolves and which components of the temporal hypergraph persist and 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.