Strong Collapse of Random Simplicial Complexes

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.