有向网络的离散里奇曲率
Discrete Ricci curvatures for directed networks
摘要：
网络科学的一个目标是复杂网络的几何特征。在这个方向上，我们最近介绍了福尔曼的离散化的里奇曲率的境界无向网络。在不同的模型和真实世界的无向网络中，对这种以边缘为中心的网络度量Forman-里奇曲率的研究表明，曲率度量捕获了无向复杂网络组织的几个方面。然而，许多重要的现实世界中的网络本质上是有向的，并且无向网络的Forman-里奇曲率的定义不适合于分析这样的有向网络。因此，我们在这里推广的Forman-里奇曲率的无向网络的情况下，有向网络。有向边的Forman-里奇曲率的简单数学公式优雅地结合了顶点权重、边权重和边方向。此外，我们还比较了Forman-里奇曲率与里奇曲率的另一个离散概念，即公认的Ollivier-里奇曲率对有向网络的适应性。然而，上面提到的两个曲率测量不考虑顶点之间的高阶相关性。为此，我们调整了福尔曼的里奇曲率的原始定义，以说明有向单纯复形，并探讨了这种新的，增强型的福尔曼-里奇曲率，在有向复杂网络的潜力。
英文摘要：
A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of undirected complex networks. However, many important real-world networks are inherently directed in nature, and the definition of the Forman-Ricci curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman-Ricci curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman-Ricci curvature of a directed edge elegantly incorporates vertex weights, edge weights and edge direction. Furthermore we have compared the Forman-Ricci curvature with the adaptation to directed networks of another discrete notion of Ricci curvature, namely, the well established Ollivier-Ricci curvature. However, the two above-mentioned curvature measures do not account for higher-order correlations between vertices. To this end, we adjusted Forman's original definition of Ricci curvature to account for directed simplicial complexes and also explored the potential of this new, augmented type of Forman-Ricci curvature, in directed complex networks.
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