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This paper proposes a flexible framework for inferring large-scaletime-varying and time-lagged correlation networks from multivariate orhigh-dimensional non-stationary time series with piecewise smooth trends. Builton a novel and unified multiple-testing procedure of time-laggedcross-correlation functions with a fixed or diverging number of lags, ourmethod can accurately disclose flexible time-varying network structuresassociated with complex functional structures at all time points. We broadenthe applicability of our method to the structure breaks by developingdifference-based nonparametric estimators of cross-correlations, achieveaccurate family-wise error control via a bootstrap-assisted procedure adaptiveto the complex temporal dynamics, and enhance the probability of recovering thetime-varying network structures using a new uniform variance reductiontechnique. We prove the asymptotic validity of the proposed method anddemonstrate its effectiveness in finite samples through simulation studies andempirical applications.

Time-varying correlation network analysis of non-stationary multivariate time series with complex trends