  The mixed-membership stochastic block model (MMSBM) is a common model for
social networks. Given an $n$-node symmetric network generated from a
$K$-community MMSBM, we would like to test $K=1$ versus $K>1$. We first study
the degree-based $\chi^2$ test and the orthodox Signed Quadrilateral (oSQ)
test. These two statistics estimate an order-2 polynomial and an order-4
polynomial of a "signal" matrix, respectively. We derive the asymptotic null
distribution and power for both tests. However, for each test, there exists a
parameter regime where its power is unsatisfactory. It motivates us to propose
a power enhancement (PE) test to combine the strengths of both tests. We show
that the PE test has a tractable null distribution and improves the power of
both tests. To assess the optimality of PE, we consider a randomized setting,
where the $n$ membership vectors are independently drawn from a distribution on
the standard simplex. We show that the success of global testing is governed by
a quantity $\beta_n(K,P,h)$, which depends on the community structure matrix
$P$ and the mean vector $h$ of memberships. For each given $(K, P, h)$, a test
is called $\textit{ optimal}$ if it distinguishes two hypotheses when
$\beta_n(K, P,h)\to\infty$. A test is called $\textit{optimally adaptive}$ if
it is optimal for all $(K, P, h)$. We show that the PE test is optimally
adaptive, while many existing tests are only optimal for some particular $(K,
P, h)$, hence, not optimally adaptive.
