Strong Collapse of Random Simplicial Complexes

TL;DR

This paper investigates the behavior of strong collapse processes on Erdős-Rényi random clique complexes, providing the first theoretical analysis of how these collapses reduce complex size while preserving homotopy.

Contribution

It offers the first theoretical results on the asymptotic size of the core after strong collapses in random simplicial complexes, specifically Erdős-Rényi clique complexes.

Findings

Remaining core size is asymptotically proportional to (1-γ)(1-cγ)n

Identifies the fixed point γ of the exponential function related to collapse behavior

Provides asymptotic almost sure results for the collapse process

Abstract

The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational topology researchers, owing to its empirically observed usefulness in simplification and size-reduction of the size of simplicial complexes while preserving the homotopy class. We consider the strong collapse process on random simplicial complexes. For the Erd\H{o}s-R\'enyi random clique complex $X(n,c/n)$ on $n$ vertices with edge probability $c/n$ with $c>1$, we show that after any maximal sequence of strong collapses the remaining subcomplex, or \emph{core} must have $(1-\gamma)(1-c\gamma) n+o(n)$ vertices asymptotically almost surely (a.a.s.), where $\gamma$ is the least non-negative fixed point of the function $f(x) = \exp\left(-c(1-x)\right)$ in the range $(0,1)$. These are the first theoretical results proved for strong collapses on random (or non-random) simplicial complexes.