We present an approach for tracking fast spatiotemporal cortical dynamics in which we combine white matter connectivity data with source-projected electroencephalographic (EEG) data. We employ the mathematical framework of graph signal processing in order to derive the Fourier modes of the brain structural connectivity graph, or "network harmonics". These network harmonics are naturally ordered by smoothness. Smoothness in this context can be understood as the amount of variation along the cortex, leading to a multi-scale representation of brain connectivity. We demonstrate that network harmonics provide a sparse representation of the EEG signal, where, at certain times, the smoothest 15 network harmonics capture 90% of the signal power. This suggests that network harmonics are functionally meaningful, which we demonstrate by using them as a basis for the functional EEG data recorded from a face detection task. There, only 13 network harmonics are sufficient to track the large-scale cortical activity during the processing of the stimuli with a 50 ms resolution, reproducing well-known activity in the fusiform face area as well as revealing co-activation patterns in somatosensory/motor and frontal cortices that an unconstrained ROI-by-ROI analysis fails to capture. The proposed approach is simple and fast, provides a means of integration of multimodal datasets, and is tied to a theoretical framework in mathematics and physics. Thus, network harmonics point towards promising research directions both theoretically - for example in exploring the relationship between structure and function in the brain - and practically - for example for network tracking in different tasks and groups of individuals, such as patients.