High-Dimensional Robust Structure Learning of Ising Models on Sparse Random Graphs

This paper considers structure learning of ferromagnetic Ising models Markov on sparse ErdősRényi random graphs with constant average degree c > 0. We propose simple, local and robust algorithms and analyze their performances in the regime of correlation decay, i.e., when c tanhJmax < 1 (where Jmax is the maximum inverse temperature in the model). The algorithms are robust because (i) they do not depend upon the specific model parameters such as the average degree and (ii) they provide guaranteed performance for a large class of sparse n-node Erdős-Rényi random graphs. We prove that a structure learning algorithm based on a set of conditional mutual information tests is consistent in high-dimensions throughout the regime of correlation decay provided the number of samples scales as ω(logn). A simpler algorithm based on correlation thresholding outputs a graph with a constant edit distance to the original graph when there is correlation decay, and the sample complexity is Ω(log n). Under a more stringent condition on the inverse temperatures (2 tanh Jmax < tanh Jmin), correlation thresholding is also shown to be consistent for structure learning. Finally, we show that Ω(c logn) samples is in fact necessary for consistent reconstruction by any algorithm. Thus, we establish that consistent structure estimation is possible with almost order-optimal sample complexity throughout the regime of correlation decay.