The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. There exist various types of such Ricci curvatures, capturing different aspects of network geometry. In the present work, we investigate Bakry-\'Emery-Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Random and small-world networks exhibit a narrow curvature distribution whereas other classes and most of the real-world networks possess a wide curvature distribution. When we compare Bakry-\'Emery-Ricci curvature with two other discrete notions of Ricci-curvature, Forman-Ricci and Ollivier-Ricci curvature for both model and real-world networks, we observe a high positive correlation between Bakry-\'Emery-Ricci and both Forman-Ricci and Ollivier-Ricci curvature, and in particular with the augmented version of Forman-Ricci curvature. Bakry-\'Emery-Ricci curvature also exhibits a high negative correlation with the vertex centrality measure and degree for most of the model and real-world networks. However, it does not correlate with the clustering coefficient. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. The computational time for Bakry-\'Emery-Ricci curvature is shorter than that required for Ollivier-Ricci curvature but higher than for Augmented Forman-Ricci curvature.

翻译：借助几何工具（例如通过所谓Ricci曲率的统计量）对复杂网络进行表征，是网络科学中一项成熟的技术。现有多种Ricci曲率类型，它们捕捉网络几何的不同方面。本文研究Bakry-Émery-Ricci曲率——一种在几何学中已得到广泛研究但尚未应用于网络的离散Ricci曲率概念。我们通过标准人工网络类别和选定的实证网络，探究该曲率的统计量与其他曲率的异同、它与其他重要网络测度的相关性，以及它能揭示的底层网络信息。我们发现，大多数顶点通常具有负曲率。随机网络和小世界网络呈现窄曲率分布，而其他网络类别及大多数真实网络则具有宽曲率分布。当我们将Bakry-Émery-Ricci曲率与其他两种离散Ricci曲率概念——Forman-Ricci曲率和Ollivier-Ricci曲率——在模型网络和真实网络上进行比较时，观察到Bakry-Émery-Ricci曲率与Forman-Ricci曲率及Ollivier-Ricci曲率（尤其是增强版Forman-Ricci曲率）呈高度正相关。此外，对于大多数模型网络和真实网络，Bakry-Émery-Ricci曲率与顶点中心性测度和度数呈高度负相关，但与聚类系数不相关。我们还研究了具有高度负曲率值的顶点在维持网络通信中的重要性。Bakry-Émery-Ricci曲率的计算时间短于Ollivier-Ricci曲率，但长于增强版Forman-Ricci曲率。