Modularity proposed by Newman and Girvan is the most commonly used measure when the nodes of a network are grouped into internally tightly and externally loosely connected communities. However, some drawbacks have been pointed out, among which is resolution limit degeneracy: being inclined to leave small communities unidentified. To overcome this drawback, Li et al. have proposed a new measure called modularity density. In this paper, we propose an equivalent formulation of the modularity density maximization as a variant of semidefinite programming, and demonstrate that its relaxation problem provides a good upper bound on the optimal modularity density. We also propose a lower bounding algorithm based on a combination of spectral heuristics and dynamic programming.
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