Collapsibility of Random Clique Complexes

TL;DR

This paper establishes conditions under which random clique complexes can be collapsed to lower dimensions and identifies thresholds for such collapsibility in sparse random models.

Contribution

It provides a new sufficient condition for collapsibility of clique complexes and determines probabilistic thresholds in Erdős–Rényi models.

Findings

Clique complexes are collapsible under specific degree conditions.

Thresholds for $(k+1)$-collapsibility depend on the probability parameter $p=n^{-eta}$.

High probability of collapsibility in sparse regimes for fixed $k$.

Abstract

We prove a sufficient condition for a finite clique complex to collapse to a $k$-dimensional complex, and use this to exhibit thresholds for $(k+1)$-collapsibility in a sparse random clique complex. In particular, if every strongly connected, pure $(k+1)$-dimensional subcomplex of a clique complex $X$ has a vertex of degree at most $2k+1$, then $X$ is $(k+1)$-collapsible. In the random model $X(n,p)$ of clique complexes of an Erd\H{o}s--R\'{e}nyi random graph $G(n,p)$, we then show that for any fixed $k\geq 0$, if $p=n^{-\alpha}$ for fixed $1/(k+1) < \alpha < 1/k$, then a clique complex $X\overset{dist}{=} X(n,p)$ is $(k+1)$-collapsible with high probability.