  This paper proposes a flexible framework for inferring large-scale
time-varying and time-lagged correlation networks from multivariate or
high-dimensional non-stationary time series with piecewise smooth trends. Built
on a novel and unified multiple-testing procedure of time-lagged
cross-correlation functions with a fixed or diverging number of lags, our
method can accurately disclose flexible time-varying network structures
associated with complex functional structures at all time points. We broaden
the applicability of our method to the structure breaks by developing
difference-based nonparametric estimators of cross-correlations, achieve
accurate family-wise error control via a bootstrap-assisted procedure adaptive
to the complex temporal dynamics, and enhance the probability of recovering the
time-varying network structures using a new uniform variance reduction
technique. We prove the asymptotic validity of the proposed method and
demonstrate its effectiveness in finite samples through simulation studies and
empirical applications.
