Reformulation of a model for hierarchical divisive graph modularity maximization

Finding clusters, or communities, in a graph, or network is a very important problem which arises in many domains. Several models were proposed for its solution. One of the most studied and exploited is the maximization of the so called modularity, which represents the sum over all communities of the fraction of edges within these communities minus the expected fraction of such edges in a random graph with the same distribution of degrees. As this problem is NP-hard, a few non-polynomial algorithms and a large number of heuristics were proposed in order to find respectively optimal or high modularity partitions for a given graph. We focus on one of these heuristics, namely a divisive hierarchical method, which works by recursively splitting a cluster into two new clusters in an optimal way. This splitting step is performed by solving a convex quadratic program. We propose a compact reformulation of such model, using change of variables, expansion of integers in powers of two and symmetry breaking constraints. The resolution time is reduced by a factor up to 10 with respect to the one obtained with the original formulation.