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

Community detection is a classic problem in network science with extensive applications in various fields. The most commonly used methods are the algorithms designed to maximize a utility function, modularity, across different ways that a network can be partitioned into communities. Despite their name and design philosophy, current modularity maximization algorithms generally fail to maximize modularity or guarantee any proximity to an optimal solution. We propose the Bayan algorithm which, unlike the existing methods, returns network partitions with a guarantee of either optimality or proximity to an optimal solution. At the core of the Bayan algorithm is a branch-and-cut scheme that solves a sparse integer programming formulation of the modularity maximization problem to optimality or approximate it within a factor. We analyze the performance of Bayan against 22 existing algorithms using synthetic and real networks. Through extensive experiments, we demonstrate Bayan's distinctive capabilities not only in maximizing modularity, but more importantly in accurate retrieval of ground-truth communities. Bayan's comparative level of performance remains stable over variations in the amount of noise in the data (graph) generation process. The performance of Bayan as an exact modularity maximization algorithm also reveals the theoretical capability limits of maximum-modularity partitions in accurate retrieval of communities. Overall our analysis points to Bayan as a suitable choice for a methodologically grounded detection of communities through exact (approximate) maximization of modularity in networks with up to $\sim10^3$ edges (and larger networks). Prospective advances in graph optimization and integer programming can push these limits further.