  The stochastic block model (SBM) is a popular framework for studying
community detection in networks. This model is limited by the assumption that
all nodes in the same community are statistically equivalent and have equal
expected degrees. The degree-corrected stochastic block model (DCSBM) is a
natural extension of SBM that allows for degree heterogeneity within
communities. This paper proposes a convexified modularity maximization approach
for estimating the hidden communities under DCSBM. Our approach is based on a
convex programming relaxation of the classical (generalized) modularity
maximization formulation, followed by a novel doubly-weighted $ \ell_1 $-norm $
k $-median procedure. We establish non-asymptotic theoretical guarantees for
both approximate clustering and perfect clustering. Our approximate clustering
results are insensitive to the minimum degree, and hold even in sparse regime
with bounded average degrees. In the special case of SBM, these theoretical
results match the best-known performance guarantees of computationally feasible
algorithms. Numerically, we provide an efficient implementation of our
algorithm, which is applied to both synthetic and real-world networks.
Experiment results show that our method enjoys competitive performance compared
to the state of the art in the literature.
