This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks' shapes that are missed by traditional connectivity and centrality graph statistics.

翻译：本文提出了一种利用拓扑数据分析领域的Z字形持续性工具研究时序网络的框架。该方法具有普适性，适用于各类时变图结构。例如，这些图可能对应于边权重随时间函数变化的网络系统模型，也可能代表复杂动力系统的时间序列。我们通过单纯复形来表征时序网络的快照，进而运用Z字形持续性进行分析。我们展示了该方法在动态网络中的两个应用：英国交通网络中多时间尺度（如日周期与周周期）通勤趋势分析，以及基于时序序数划分网络表征的动力系统中由间歇现象引发的周期/混沌转换检测。研究结果表明，所生成的零维与一维Z字形持续性图能够捕捉传统连通性与中心性图统计量无法识别的网络拓扑形态变化。