Computational techniques for network clustering identification are critical to several application domains. Recently, Modularity Maximization and Modularity Density Maximization have become two of the main techniques that provide computational methods to identify network clusterings. Therefore, understanding their differences and common characteristics is fundamental to decide which one is best suited for a given application. Several heuristics and exact methods have been developed for both Modularity Maximization and Modularity Density Maximization problems. Unfortunately, no structured methodological comparison between the two techniques has been proposed yet. This paper reports a ground truth contest between both optimization problems. We do so aiming to compare their exact solutions and the results of heuristics inspired in these problems. In our analysis, we use branch-and-price exact methods which apply the best-known column generation procedures. The heuristic methods obtain the highest objective function scores and find solutions for networks with hundreds of thousands of nodes. Our experiments suggest that Modularity Density Maximization yields the best results over the tested networks. The experiments also show the behavior and importance of the quantitative factor of the Modularity Density Maximization objective function.

中文翻译：

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模块化最大化和模块化密度最大化之间的真实竞赛

用于网络聚类识别的计算技术对多个应用领域至关重要。最近，模块化最大化和模块化密度最大化已经成为提供计算方法来识别网络聚类的两种主要技术。因此，了解它们的差异和共同特征是决定哪一种最适合给定应用的基础。已经为模块化最大化和模块化密度最大化问题开发了几种启发式和精确方法。不幸的是，尚未提出两种技术之间的结构化方法比较。本文报告了两个优化问题之间的真实情况竞赛。我们这样做的目的是比较他们的确切解决方案和启发式在这些问题中的结果。在我们的分析中，我们使用应用最知名的列生成程序的分支和价格精确方法。启发式方法获得最高的目标函数分数，并为具有数十万个节点的网络找到解决方案。我们的实验表明，模块化密度最大化在测试网络上产生了最好的结果。实验还显示了模块化密度最大化目标函数的定量因素的行为和重要性。