Information-theoretic limits of graphical model selection in high dimensions

The problem of graphical model selection is to correctly estimate the graph structure of a Markov random field given samples from the underlying distribution. We analyze the information-theoretic limitations of this problem under high-dimensional scaling, in which the graph size p and the number of edges k (or the maximum degree d) are allowed to increase to infinity as a function of the sample size n. For pairwise binary Markov random fields, we derive both necessary and sufficient conditions on the scaling of the triplet (n, p, k) (or the triplet (n, p, d)) for asympotically reliable reocovery of the graph structure.