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The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model togenerate random graphs with community structure given an expected degreesequence. The standard approach of community detection based on the DCSBM is tosearch for the model parameters that are the most likely to have produced theobserved network data through maximum likelihood estimation (MLE). Currenttechniques for the MLE problem are heuristics, and therefore do not guaranteeconvergence to the optimum. We present mathematical programming formulationsand exact solution methods that can provably find the model parameters andcommunity assignments of maximum likelihood given an observed graph. We comparethese exact methods with classical heuristic algorithms based onexpectation-maximization (EM). The solutions given by exact methods give us aprincipled way of measuring the experimental performance of classicalheuristics and comparing different variations thereof.

Community Detection in the Stochastic Block Model by Mixed Integer Programming