  How to detect a small community in a large network is an interesting problem,
including clique detection as a special case, where a naive degree-based
$\chi^2$-test was shown to be powerful in the presence of an Erd\H{o}s-Renyi
background. Using Sinkhorn's theorem, we show that the signal captured by the
$\chi^2$-test may be a modeling artifact, and it may disappear once we replace
the Erd\H{o}s-Renyi model by a broader network model. We show that the recent
SgnQ test is more appropriate for such a setting. The test is optimal in
detecting communities with sizes comparable to the whole network, but has never
been studied for our setting, which is substantially different and more
challenging. Using a degree-corrected block model (DCBM), we establish phase
transitions of this testing problem concerning the size of the small community
and the edge densities in small and large communities. When the size of the
small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it
attains the computational lower bound (CLB), the information lower bound for
methods allowing polynomial computation time. When the size of the small
community is smaller than $\sqrt{n}$, we establish the parameter regime where
the SgnQ test has full power and make some conjectures of the CLB. We also
study the classical information lower bound (LB) and show that there is always
a gap between the CLB and LB in our range of interest.
