Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in complex networks. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets.

翻译：Forman-Ricci曲率（FRC）是分析经验网络的一种强有力工具，其曲率值分布能够识别其他几何方法难以检测的结构信息。关键在于，FRC可捕捉图团复形或Vietoris-Rips复形的高阶结构信息，而这正是其他替代方法不易获取的。然而，现有FRC平台的计算成本过高。因此，本文提出了一种高效的集合论形式，用于计算复杂网络中的高阶FRC。值得注意的是，我们的集合论表示揭示了之前的计算瓶颈，并加速了FRC的计算。最后，我们提供了伪代码、名为FastForman的软件实现，以及与替代实现的基准比较。我们预计FastForman将用于高维复杂数据集的拓扑与几何数据分析。