Optimality Of Community Structure In Complex Networks

Community detection is one of the pivotal tools for discovering the structure of complex networks. Majority of community detection methods rely on optimization of certain quality functions characterizing the proposed community structure. Perhaps, the most commonly used of those quality functions is modularity. Many heuristics are claimed to be efficient in modularity maximization, which is usually justified in relative terms through comparison of their outcomes with those provided by other known algorithms. However as all the approaches are heuristics, while the complete brute-force is not feasible, there is no known way to understand if the obtained partitioning is really the optimal one. In this article we address the modularity maximization problem from the other side --- finding an upper-bound estimate for the possible modularity values within a given network, allowing to better evaluate suggested community structures. Moreover, in some cases when then upper bound estimate meets the actually obtained modularity score, it provides a proof that the suggested community structure is indeed the optimal one. We propose an efficient algorithm for building such an upper-bound estimate and illustrate its usage on the examples of well-known classical and synthetic networks, being able to prove the optimality of the existing partitioning for some of the networks including well-known Zachary's Karate Club.