Very recently, the locally stationary wavelet framework has provided a means to describe the dependencies of co-varying time-series over a range of multiple scale levels. However, describing the many interactions between data-streams at different scale levels with only finite data poses some serious statistical estimation challenges. We illustrate that existing approaches suffer from large variance and are sometimes difficult to interpret. We here propose a sparsity-aware estimator which furnishes a set of multiresolution, dynamic graphs that describe how the dependency structure of the variables evolves through time and over multiple levels of scale. We show that the regulariser mitigates the variance and that, since the inference is performed using convex optimisation, it converges quickly to a global optima and scales well with respect to samples and nodes. Basic properties of the new method are established on simulated data. The method is applied to inferring dependency structure in multivariate EEG data-sets during epileptic seizures where it reveals evidence of band-limited dependency structure.