Modularity maximization in networks by variable neighborhood search

Finding communities, or clusters, in networks, or graphs, has been the subject of intense studies in the last ten years. The most used criterion for that purpose, despite some recent criticism, is modularity maximization, proposed by Newman and Girvan. It consists in maximizing the sum for all clusters of the number of inner edges minus the expected number of inner edges assuming the same distribution of degrees. Numerous heuristics, as well as a few exact algorithms have been proposed to maximize modularity. We apply the Variable Neighborhood Search metaheuristic to that problem. The resulting Variable Neighborhood Decomposition Search heuristic is applied to the Clustering problems of the 10 DIMACS Implementation Challenge. The optimal solution is obtained by the heuristic whenever it is known, and otherwise the best solutions from the literature are improved, except in the case of three instances for which available memory was too small.