Estimation of Hub Structures in Individual and Multiple Gaussian Graphical Models

Due to recent advances in technology, many scientific disciplines have experienced a rapid growth in the generation of big data. These datasets can often be analyzed to recover complex relationships among large sets of variables, such as the presence of correlation or a graphical model dependence structure. The recovery of these variable dependence structures can provide further understanding of scientific phenomena and lead to important discoveries. Furthermore, it is vital to develop statistical methods that can not only detect structure for a single population, but also infer the presence of common and individual structure across multiple populations of interest. In this dissertation, we focus on efficient structure estimation of high-dimensional graphical models and correlation matrices. First, we study the problem of estimating hub variables in Gaussian graphical models (GGMs), which refers to detecting nodes with a high degree of connectivity compared to other nodes. To this end, we show a novel connection between the presence of hub variables in GGMs and the spectral decomposition of the precision matrix associated with the data. By applying this characterization, we can estimate the presence of hub variables in a GGM only by studying the eigenvalues and eigenvectors of the associated sample covariance matrix. We further extend our method for the detection of hub variables that are common across multiple different GGMs as well. Finally, we propose a permutation resampling method for testing the equality of correlation matrices originating from multiple populations. Our method is shown to be effective regardless of whether the difference in the correlation matrices is sparse or dense.