Graph clusterings with overlaps: Adapted quality indices and a generation model

Many real-world networks have a structure of overlapping cohesive groups. In order to uncover this structure several clustering algorithms have been developed. In this paper, we focus on the evaluation of these algorithms. Quality measures are commonly used for this purpose and provide a means to assess the quality of a derived cluster structure. Currently, there are too few measures for graph clusterings with overlaps available that would enable a meaningful evaluation, even though many well studied crisp quality measures exist. In order to expand the pool of overlapping measures we propose three methods to adapt existing crisp quality measures so that they can handle graph overlaps appropriately. We demonstrate our methods on the well known measures Density, Modularity and Conductance. We also propose an enhancement of an existing modularity measure for networks with overlapping structure. We analyse the proposed quality indices using experiments on artificial graphs that possess overlapping structure. For this evaluation, we apply a graph generation model to create clustered graphs with overlaps that are similar to real-world networks, i.e., their node degree and cluster size distribution follow a power law.