Sample complexity of determining structures of graphical models

We consider discovering the graph structure of a pairwise Markov random field (MRF) on p binary random variables using n samples from the underlying MRF distribution. We analyze the information-theoretic limitations of this problem under high-dimensional scaling, when the number of connections of each variable in the underlying MRF is bounded by d.We derive both necessary and sufficient conditions on the scaling of the triplet (n, p, d) for asympotically reliable recovery of the graph structure.