  This paper surveys recent theoretical advances in convex optimization
approaches for community detection. We introduce some important theoretical
techniques and results for establishing the consistency of convex community
detection under various statistical models. In particular, we discuss the basic
techniques based on the primal and dual analysis. We also present results that
demonstrate several distinctive advantages of convex community detection,
including robustness against outlier nodes, consistency under weak
assortativity, and adaptivity to heterogeneous degrees.
  This survey is not intended to be a complete overview of the vast literature
on this fast-growing topic. Instead, we aim to provide a big picture of the
remarkable recent development in this area and to make the survey accessible to
a broad audience. We hope that this expository article can serve as an
introductory guide for readers who are interested in using, designing, and
analyzing convex relaxation methods in network analysis.
