  This paper is concerned with the estimation of time-varying networks for
high-dimensional nonstationary time series. Two types of dynamic behaviors are
considered: structural breaks (i.e., abrupt change points) and smooth changes.
To simultaneously handle these two types of time-varying features, a two-step
approach is proposed: multiple change point locations are first identified
based on comparing the difference between the localized averages on sample
covariance matrices, and then graph supports are recovered based on a
kernelized time-varying constrained $L_1$-minimization for inverse matrix
estimation (CLIME) estimator on each segment. We derive the rates of
convergence for estimating the change points and precision matrices under mild
moment and dependence conditions. In particular, we show that this two-step
approach is consistent in estimating the change points and the piecewise smooth
precision matrix function, under certain high-dimensional scaling limit. The
method is applied to the analysis of network structure of the S\&P 500 index
between 2003 and 2008.
