  We propose and analyze the problems of \textit{community goodness-of-fit and
two-sample testing} for stochastic block models (SBM), where changes arise due
to modification in community memberships of nodes. Motivated by practical
applications, we consider the challenging sparse regime, where expected node
degrees are constant, and the inter-community mean degree ($b$) scales
proportionally to intra-community mean degree ($a$). Prior work has sharply
characterized partial or full community recovery in terms of a "signal-to-noise
ratio" ($\mathrm{SNR}$) based on $a$ and $b$. For both problems, we propose
computationally-efficient tests that can succeed far beyond the regime where
recovery of community membership is even possible. Overall, for large changes,
$s \gg \sqrt{n}$, we need only $\mathrm{SNR}= O(1)$ whereas a na\"ive test
based on community recovery with $O(s)$ errors requires $\mathrm{SNR}=
\Theta(\log n)$. Conversely, in the small change regime, $s \ll \sqrt{n}$, via
an information-theoretic lower bound, we show that, surprisingly, no algorithm
can do better than the na\"ive algorithm that first estimates the community up
to $O(s)$ errors and then detects changes. We validate these phenomena
numerically on SBMs and on real-world datasets as well as Markov Random Fields
where we only observe node data rather than the existence of links.
