Modeling and Simulating Dependence in Networks Using Topological Data Analysis

Abstract: Topological data analysis (TDA) approaches are becoming increasingly popular for studying the dependence patterns in multivariate time series data. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes, which can be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Simulations are crucial for evaluating the performance of proposed TDA methods and testing procedures as well as for creating computation-based confidence intervals. To our knowledge, there are no methods that simulate multivariate time series data with specific and manually imposed connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.