Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex

The topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since a single parameter usually governs the randomness in these models. With this in mind, we focus here on the topology of the recently proposed multi-parameter random simplicial complex. In particular, we introduce a dynamic variant of this model and look at how its topology evolves. In this dynamic setup, the temporal evolution of simplices is determined by stationary and possibly non-Markovian processes with a renewal structure. Special cases of this setup include the dynamic versions of the clique complex and the Linial–Meshulam complex. Our key result concerns the regime where the face-count of a particular dimension dominates. We show that the Betti number corresponding to this dimension and the Euler characteristic satisfy a functional strong law of large numbers and a functional central limit theorem. Surprisingly, in the latter result, the limiting process depends only upon the dynamics in the smallest non-trivial dimension.