Signal detection in degree corrected ERGMs

In this paper, we study sparse signal detection problems in "degree corrected" Exponential Random Graph Models (ERGMs). We study the performance of two tests based on conditionally centered sum of degrees/maximum of degrees, for a wide class of such ERGMs. The performance of these tests match the performance of corresponding uncentered tests in the β model (Ann. Statist. 46 (2018) 1288–1317). Focusing on the degree corrected two star ERGM, we show that improved detection is possible at "criticality" using a test based on (unconditional) sum of degrees. In this setting we provide matching lower bounds in all parameter regimes, which is based on correlations estimates between degrees under the alternative, and is of possible independent interest.