  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 the conditionally centered sum of degrees and conditionally centered
maximum of degrees, for a wide class of such ERGMs. The performance of these
tests match the performance of the corresponding uncentered tests in the
$\beta$ model. 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 of possible independent interest.
