  The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model to
generate random graphs with community structure given an expected degree
sequence. The standard approach of community detection based on the DCSBM is to
search for the model parameters that are the most likely to have produced the
observed network data through maximum likelihood estimation (MLE). Current
techniques for the MLE problem are heuristics, and therefore do not guarantee
convergence to the optimum. We present mathematical programming formulations
and exact solution methods that can provably find the model parameters and
community assignments of maximum likelihood given an observed graph. We compare
these exact methods with classical heuristic algorithms based on
expectation-maximization (EM). The solutions given by exact methods give us a
principled way of measuring the experimental performance of classical
heuristics and comparing different variations thereof.
