We define localized partial correlation and Granger causality graphs for possibly nonstationary and non-Gaussian multivariate time series processes using wavelet-based methods. The nodes denote the component processes and the edges describe the pairwise local conditional dependence, after accounting for contemporaneous and lagged influences of the remaining variables. Local linear dependence is characterized in our model by the partial coherence measure defined in time-frequency domain. For empirical data, we propose a Wavelet-penalization technique to estimate the graph structure at multiple scale and time points in additive noise settings. The methodology is applied to realized volatilities for the largest equity indexes worldwide.