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

Abstract

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. c

Random Simplicial Complexes: Models and Phenomena

“…The limiting processes are shown to be Ornstein-Uhlenbeck (Gaussian) processes. These results were further extended to the multiparameter complex [110].…”

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

“…Distributional limit theorems for the multi-parameter simplicial complex K(n; p) were obtained in Owada et al (2021). These results are obtained under the assumption (4.1)…”

“…this introduces a minor unimportant abuse of notation by conflating the dimension of the simplex with the largest dimension of an entire complex. Counting the simplices at the critical dimension is particularly important since their numbers largely determine the behaviour of the Euler characteristic of K(n; p); Thoppe et al ( 2016); Owada et al (2021). Let σ k be a simplex of dimension k satisfying (4.2).…”

“…Kahle and Meckes (2013); Thoppe et al (2016); Fowler (2019); Kanazawa (2022). Functional limit theorems for a dynamic version of the multiparameter model have been established in Owada et al (2021).…”

“…In Section 3, we address general upper tail large deviation problem for subcomplex counts. Section 4 specializes to the large deviation problem for the number of simplices at the dominating dimension and below, under the setup of Owada et al (2021). Finally, Section 5 discusses the upper tail large deviations for Betti numbers at the dominating dimension.…”