Abstract: In this talk, we discuss optimal hypothesis testing for distinguishing a stochastic block model from an Erdos--Renyi random graph when the average degree grows to infinity with the graph size. We show that linear spectral statistics based on Chebyshev polynomials of the adjacency matrix can approximate signed cycles of growing lengths when the graph is sufficiently dense. The signed cycles have been shown by Banerjee (2018) to determine the likelihood ratio statistic asymptotically. In this way one achieves sharp…

Abstract: Inference in the framework of Ising models has received significant attention in Statistics and Machine Learning in recent years. In this talk we study joint estimation of the inverse temperature parameter β, and the magnetization parameter B, given one realization from the Ising model, under the assumption that the underlying graph of the Ising model is completely specified. We show that if the graph is either irregular or sparse, then both the parameters can be estimated at rate n−1/2…