The SmartGrid (SG) is a complex system connecting physical components (e.g., human, weather, power plants) and logical components (e.g., control algorithms, communication infrastructure, protocols). The large number of components and the interactions between the individual components induce an extremely intricate behavior of the overall system. Detecting anomalies in the behavior of the system requires a large number of observations and is unpractical. A novel learning and estimation framework to analyze stochastic processes over graphs associated with SG systems is proposed. The critical observation behind the proposed framework in that these systems induce an underlying sparse structure which enables dimension reduction via compressed sensing-like schemes. Numerical results show that the compression approach proposed herein reduces by orders of magnitude the number of observations required to detect an anomalous behavior of the SG.