Modified Modularity Density Maximization and Density Ratio Heuristic

Network science is considered as an interdisciplinary research field engaging engineering, social, natural, and computer sciences. Most networks nestle vertices organized in groups called communities, modules or clusters. Communities are groups of vertices that perhaps partake similar features and/or function common roles within a graph. Modularity maximization objective is employed in community detection algorithms. However, modularity maximization solution has practical problems such as resolution limit and frailty. Recently, an alternative clustering measure known as modularity density has been developed to address the resolution limit of modularity maximization. Modularity Density Maximization (MDM) aims to reduce the out of cluster links. The less out connections improve the objective function. In this research, the out connections are perceived as distances. Thus, we propose a Modified Modularity Density Maximization (MMDM) as we consider minimizing the deepest out connection instead of minimizing the out links. Modified Modularity Density Maximization (MMDM) is formulated as a Mixed Integer Linear Programming (MILP). GAMS software is used to solve the model and the obtained results are compared with MDM using internal cluster validation approach. A novel heuristic clustering algorithm named Density Ratio (DR) Heuristic. DR is proposed to solve larger data sets that cannot be solved by MILP or take very long time to solve. The heuristic is applied on both MMDM and MDM approaches and the obtained results are compared using internal cluster validation approach. MMDM finds higher values compared to MDM based on internal validation for some datasets. The density ratio heuristic finds global optimal or near optimal solutions with the datasets employed for the MILP model.