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.