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.

翻译：本文研究了时间超图的拓扑性质。超图作为图的高维推广，能够捕捉多路连接，已成为网络科学的重要组成部分。超图的一个常见用途是将事件建模为超边，其中事件可涉及多个元素作为节点。这提供了对事件更完整的描述，不受标准图对偶连接的限制。然而，事件通常附有作为时间间隔的时间信息，用于指示事件发生的时间段。因此，时间超图应运而生，它能够同时准确捕捉事件的时间信息及其多路连接。研究这些时间超图的常用工具通常通过滑动窗口过程中采样的快照的汇总统计来捕捉底层动态的变化，但这些工具既无法刻画超图结构随时间的演化，也无法揭示对底层系统有影响力的持久成分。为满足这一需求，我们利用拓扑数据分析（TDA）领域的锯齿持久性来研究随时间演化的超图的拓扑结构变化。我们将该分析流程应用于网络安全和社交网络数据集，展示了其时间超图拓扑结构的变化，并解释了这些变化如何用于理解底层动态。