We have performed an empirical comparison of two distinct notions of discrete
Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and
Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci
curvature were developed based on different properties of the classical smooth
notion, and thus, the two notions shed light on different aspects of network
structure and behavior. Nevertheless, our extensive computational analysis in a
wide range of both model and real-world networks shows that the two
discretizations of Ricci curvature are highly correlated in many networks.
Moreover, we show that if one considers the augmented Forman-Ricci curvature
which also accounts for the two-dimensional simplicial complexes arising in
graphs, the observed correlation between the two discretizations is even
higher, especially, in real networks. Besides the potential theoretical
implications of these observations, the close relationship between the two
discretizations has practical implications whereby Forman-Ricci curvature can
be employed in place of Ollivier-Ricci curvature for faster computation in
larger real-world networks whenever coarse analysis suffices.

本研究对图或网络的两种离散 Ricci 曲率形式，即 Forman-Ricci 曲率和 Ollivier-Ricci 曲率进行了实证比较分析。在更广泛的模型和实际网络中进行了广泛的计算分析，表明了两种离散曲率在许多网络中高度相关，具有相似的结构和行为特性。此外，通过引入转化 Forman-Ricci 曲率，研究显示，两种曲率之间的相关性更高，特别是在真实网络中，可以利用 Forman-Ricci 曲率来替代 Ollivier-Ricci 曲率进行更快速的计算和分析。

复杂网络中两种离散的 Ricci 曲率的比较分析

Comparative analysis of two discretizations of Ricci curvature for  complex networks

We introduce Forman-Ricci curvature and its corresponding flow as
characteristics for complex networks attempting to extend the common approach
of node-based network analysis by edge-based characteristics. Following a
theoretical introduction and mathematical motivation, we apply the proposed
network-analytic methods to static and dynamic complex networks and compare the
results with established node-based characteristics. Our work suggests a number
of applications for data mining, including denoising and clustering of
experimental data, as well as extrapolation of network evolution.

介绍 Forman-Ricci 曲率及其对应的流作为复杂网络的特征，旨在扩展基于节点的网络分析方法。将该方法应用于静态和动态复杂网络，并与已建立的基于节点的特征进行比较，建议其在数据挖掘中的应用，包括实验数据去噪和聚类，以及网络演化的外推。