Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering, and their discretizations enable the development of network-based mathematical and computational frameworks, which are essential for large-scale data science. The Forman-Ricci curvature (FRC) - a statistical measure based on Riemannian geometry and designed for networks - is known for its high capacity for extracting geometric information from complex networks. However, extracting information from dense networks is still challenging due to the combinatorial explosion of high-order network structures. Motivated by this challenge we sought a set-theoretic representation theory for high-order network cells and FRC, as well as their associated concepts and properties, which together provide an alternative and efficient formulation for computing 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. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.

翻译：微分几何方法在数学、物理和工程等多个领域普遍存在，其离散化形式促进了基于网络的数学与计算框架的发展，这对大规模数据科学至关重要。Forman-Ricci曲率（FRC）——一种基于黎曼几何并专为网络设计的统计度量——以其从复杂网络中提取几何信息的高能力而闻名。然而，由于高阶网络结构的组合爆炸，从稠密网络中提取信息仍具挑战性。受此挑战驱动，我们探索了一种针对高阶网络单元和FRC的集合论表示理论，及其相关概念和性质，该理论共同为计算复杂网络中的高阶FRC提供了一种替代且高效的方案。我们给出了伪代码、名为FastForman的软件实现，以及与其他实现方式的基准对比。关键在于，我们的表示理论揭示了先前的计算瓶颈，并加速了FRC的计算。因此，我们的发现为需要高阶几何计算的复杂系统开辟了新的研究可能性。