Topology of random d-clique complexes

TL;DR

This paper investigates the topological properties of random d-clique complexes, establishing conditions under which certain homology groups vanish or persist, thus advancing understanding of their asymptotic behavior.

Contribution

It provides new thresholds for the vanishing and non-vanishing of homology groups in random d-clique complexes, partially answering a question by Eric Babson.

Findings

Homology groups vanish above certain thresholds.

Homology groups persist below certain thresholds.

Results depend on the parameter p = n^α.

Abstract

For a simplicial complex $X$, the $d$-clique complex $\Delta_d(X)$ is the simplicial complex having all subsets of vertices whose $(d + 1)$-subsets are contained by $X$ as its faces. We prove that if $p = n^{\alpha}$, with $\alpha < \max\{\frac{-1}{k-d +1},-\frac{d+1}{\binom{k}{d}}\}$ or $\alpha > \frac{-1}{\binom{2k+2}{d}}$, then the $k$-th reduced homology group of the random $d$-clique complex $\Delta_d(G_d(n,p))$ is asymptotically almost surely vanishing, and if $\frac{-1}{t} < \alpha < \frac{-1}{t+1}$ where $t = (\frac{(d+1)(k+1)}{\binom{(d+1)(k+1)}{d+1}-(k+1)})^{-1}$, then the $(kd + d -1)$-st reduced homology group of $\Delta_d(G_d(n,p))$ is asymptotically almost surely nonvanishing. This provides a partial answer to a question posed by Eric Babson.