Efficient Algorithms for Extracting Higher-Order Geometric Information from Complex Networks

Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering. The Forman-Ricci curvature (FRC) 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 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. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.