Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes

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.