Estimating Sparse Network Models

The statistical modeling of networks can be useful for discovering complex multivariate dependencies, especially for high-dimensional data when the sample size is relatively small. Existing methods in the literature propose methods for recovering the concentration matrix or covariance matrix when the associate graph is sparse. In many real life applications many networks also have some nodes/variables with a high degree of connectivity. These nodes play a critical role in the functioning of the network. In this paper we formally define the class of sparse network models with ``highly leveraged" nodes and explore inference for this class of models. We propose a penalized log-likelihood method for recovering this model from data that has the same computational efficiency as current methods in the literature. We observe that our novel approach achieves its objectives by simultaneously estimating the sparse network and the highly connected nodes correctly. The methodology is applied to real life data sets and simulation experiments to illustrate the efficacy of our procedure. (This is joint work with R.Mazumder)