Multivariate Central Limit Theorems for Random Clique Complexes

TL;DR

This paper proves multivariate normal approximation theorems for key vectors in random clique complexes, advancing understanding in algebraic topology and related computational algorithms.

Contribution

It introduces a multivariate CLT for dissociated sums, extending Stein's method to analyze vectors arising in random clique complexes.

Findings

Multivariate normal approximation for critical simplex counts

Generalization of vertex degree concept in random complexes

Application of Stein's method to multivariate dissociated sums

Abstract

Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: (1) the vector of critical simplex counts attained by a lexicographical Morse matching, (2) the vector of simplex counts in the link of a fixed simplex, and (3) the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U-statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein's method.