Forman-Ricci curvature (FRC) has become a potent and powerful for analysing empirical networks, as the distribution of the curvature values can identify some structural information that is not readily detected by other schemes. For a graph, it is easy to compute, but it is not sensitive to higher-order structural information. That issue can be handled by going to the clique complex of the graph, and in fact, such complexes also emerge as Vietoris-Rips complexes in Topological Data Analysis. But then, with standard methods, it becomes much more computationally expensive. In this paper, we therefore develop a novel set-theoretic formulation for computing such high-order FRC in complex networks. We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. Crucially, our representation theory reveals previous computational bottlenecks and also accelerates the computation of FRC.

翻译：Forman-Ricci曲率（FRC）已成为分析经验网络的有效且强大的工具，因为曲率值的分布可以识别其他方法难以检测到的某些结构信息。对于图而言，其计算简单，但对高阶结构信息不敏感。这一问题可以通过将图转化为团复形来解决——事实上，这类复形在拓扑数据分析中也会以Vietoris-Rips复形的形式出现。然而，采用标准方法时，计算成本会显著增加。为此，本文提出了一种新颖的基于集合论的公式，用于计算复杂网络中的此类高阶FRC。我们提供了伪代码、名为FastForman的软件实现，并与其他替代实现进行了基准比较。关键在于，我们的表示理论揭示了以往的计算瓶颈，并加速了FRC的计算。