In this talk, Dimosthenis Pasadakis will discuss computationally efficient and accurate algorithms for the estimation of graphs from high-dimensional data, and for subsequent anomaly detection tasks. Initially, he examines a performant inverse covariance matrix estimation method, based on the sparse quadratic approximation of the L1-regularized Gaussian maximum likelihood. Its capabilities are extended to the retrieval of graphs of only non-negatively correlated variables, or equivalently, the problem of sparse adjacency estimation. Last, we consider the problem of detecting outliers in networks and leverage spectral graph partitioning principles to perform anomaly detection in a multilevel fashion. The advantages of all introduced algorithms are showcased in a series of comparative tests with the current state-of-the-art on artificial datasets, and their real-world applicability is demonstrated with numerical experiments on medical and financial data.