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Community detection is a classic problem in network science with extensiveapplications in various fields. Among numerous approaches, the most commonmethod is modularity maximization. Despite their design philosophy and wideadoption, heuristic modularity maximization algorithms rarely return an optimalpartition or anything similar. We propose a specialized algorithm, Bayan, whichreturns partitions with a guarantee of either optimality or proximity to anoptimal partition. At the core of the Bayan algorithm is a branch-and-cutscheme that solves an integer programming formulation of the modularitymaximization problem to optimality or approximate it within a factor. Wecompare Bayan against 30 alternative community detection methods usingstructurally diverse synthetic and real networks. Our results demonstrateBayan's distinctive accuracy and stability in retrieving ground-truthcommunities of standard benchmark graphs. Bayan is several times faster thanopen-source and commercial solvers for modularity maximization making itcapable of finding optimal partitions for instances that cannot be optimized byany other existing method. Overall, our assessments point to Bayan as asuitable choice for exact maximization of modularity in real networks with upto 3000 edges (in their largest connected component) and approximating maximummodularity in larger instances on ordinary computers. A Python implementationof the Bayan algorithm (the bayanpy library) is publicly available through thepackage installer for Python (pip).

The Bayan Algorithm: Detecting Communities in Networks Through Exact and Approximate Optimization of Modularity