![]() A modularity solution is a partition of disjoint nodes that maximizes an objective function which represents the difference between the internal and the expected cluster connectivity. Modularity optimization was first introduced by Newman and Girvan (2004) and uses the modular feature to determine nodes that belong to each cluster. In these methods, the modular property of a set of nodes is used to detect clusters which have a higher internal than external connection density (Fortunato, Castellano, 2012, Radicchi, Castellano, Cecconi, Loreto, Parisi, 2004, Xie, Kelley, Szymanski, 2013). The latter is expressed as edge or arc weights. Identifying communities in networks is a relevant application in different fields, e.g.: ( i) biology, to find protein complexes (Nepusz, Yu, & Paccanaro, 2012) and to map metabolic reactions (Guimera & Amaral, 2005) ( ii) health sciences, to identify functional memory used for olfactory recognition (Meunier et al., 2014) ( iii) social sciences, to recognize individuals in criminal networks (Ferrara, De Meo, Catanese, & Fiumara, 2014).įor community detection, some methods only use the graph topology and the relationship strength. Ground truth experiments in artificial random graphs were performed and suggest that our heuristics lead to better cluster detection than both CNM and Louvain. Hypothesis tests suggest that four proposed heuristics are state-of-the-art since they are scalable for hundreds of thousands of nodes for the modularity density problem, and they find the high objective value partitions for the largest instances. This feature was confirmed by an amortized complexity analysis which reveals average linear time for three of our heuristics. Our seven heuristics were tested with real graphs from the Stanford Large Network Dataset Collection and the experiments show that they are scalable. Our experiments also show that some of our heuristics surpassed the objective function value reported by iMeme-Net, Hain, and BMD- λ for some real graphs. The results suggest that our seven heuristics are faster than GAOD, iMeme-Net, HAIN, and BMD- λ modularity density heuristics. The results are also compared with CNM and Louvain, which are scalable heuristics for modularity maximization. This paper presents seven scalable heuristics for modularity density and compares them with literature results from exact mixed integer linear programming and GAOD, iMeme-Net, HAIN, and BMD- λ heuristics. Modularity density maximization is a community detection optimization problem which improves the resolution limit degeneracy of modularity maximization.
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