Emergence of Betti numbers in growing simplicial complexes: analytical solutions

A simplicial complex (SC) is a mathematical object that represents high-dimensional interactions. Here, we investigate homological percolation transitions (HPTs), which are topological deformations that occur as an SC grows. Employing basic techniques from homological algebra, we determine an HPT by the nth Betti number, which represents the number of n-dimensional holes. A previous study showed that the HPTs of successive nth Betti numbers represent the evolutionary steps of a coauthorship network, which indicate significant changes in collaboration patterns among research groups; however, these results were obtained only numerically. Here, we present analytic solutions for the properties of HPTs with n = 0 and n = 1, based on the minimal model proposed in the previous study. Because the coauthorship network is a growing network, the HPTs are of infinite order, and the order parameter has an essentially singular form but different exponent values for different Betti numbers, n = 1 and 2. (paper)