Topological similarity of time-dependent objects

Objects with time factor are important representations of data, such as time series, sequences generated by chaotic systems, and dynamic networks. This paper focuses on characterizing the topological similarity of such objects. First, we introduce time-dependent object (TDO), which is time-dependent datasets with metric structures. Second, we extend the concept of topological correlation coefficient so that it can analyze time-dependent objects. The generalized topological correlation coefficient (GTCC) can be well defined on TDO and characterize the topological similarity between datasets. Third, we analyze examples constructed from chaotic systems and temporal networks. Calculations show that there are non-trivial topological similarities in both datasets, where surrogate TDOs provide benchmark values for comparison. In particular, GTCC can identify patterns in the temporal network arising from human behavior. The analytical framework presented in this paper has the potential to be widely used in time series analysis, nonlinear dynamical systems, and dynamic networks.