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 simplicial complexes. 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. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.

翻译：Forman-Ricci曲率（FRC）是分析经验网络的强大工具，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能够捕捉图团复形或Vietoris-Rips复形中高阶结构信息，这是替代方法难以直接获取的。然而，现有FRC计算平台的计算成本过高。为此，本文提出了一种高效的集合论公式，用于计算单纯复形中的高阶FRC。值得注意的是，我们的集合论表示不仅揭示了以往的计算瓶颈，还加速了FRC的计算。最后，我们提供了伪代码、名为FastForman的软件实现，并与其他实现进行了基准对比。我们预计FastForman将应用于高维复杂数据集的拓扑与几何数据分析。此外，我们的发展为未来高效计算胞腔复形上FRC的推广奠定了基础。