A mathematical programming approach to overlapping community detection

Abstract

We propose a new optimization model to detect overlapping communities in networks. The model elaborates suggestions contained in Zhang et al. (2007), in which overlapping communities were identified through the use of a fuzzy membership function, calculated as the outcome of a mathematical programming problem. In our approach, we retain the idea of using both mathematical programming and fuzzy membership to detect overlapping communities, but we replace the fuzzy objective function proposed there with another one, based on the Newman and Girvan’s definition of modularity. Next, we formulate a new mixed-integer linear programming model to calculate optimal overlapping communities. After some computational tests, we provide some evidence that our new proposal can fix some biases of the previous model, that is, its tendency of calculating communities composed of almost all nodes. Conversely, our new model can reveal other structural properties, such as nodes or communities acting as bridges between communities. Finally, as mathematical programming can be used only for moderate size networks due to its computation time, we proposed two heuristic algorithms to solve the largest instances, that compare favourably to other methodologies.