  We consider the spectral properties of sparse stochastic block models, where
$N$ vertices are partitioned into $K$ balanced communities. Under an assumption
that the intra-community probability and inter-community probability are of
similar order, we prove a local semicircle law up to the spectral edges, with
an explicit formula on the deterministic shift of the spectral edge. We also
prove that the fluctuation of the extremal eigenvalues is given by the GOE
Tracy--Widom law after rescaling and centering the entries of sparse stochastic
block models. Applying the result to sparse stochastic block models, we
rigorously prove that there is a large gap between the outliers and the
spectral edge without centering.
