Maximal persistence in random clique complexes

Abstract

We study the persistent homology of an Erd\H{o}s--R\'enyi random clique complex filtration on $n$ vertices. Here, each edge $e$ appears at a time $p_e \in [0,1]$ chosen uniform randomly in the interval, and the \emph{persistence} of a cycle $\sigma$ is defined as $p_2 / p_1$, where $p_1$ and $p_2$ are the birth and death times of the cycle respectively. We show that for fixed $k \ge 1$, with high probability the maximal persistence of a $k$-cycle is of order roughly $n^{1/k(k+1)}$. These results are in sharp contrast with the random geometric setting where earlier work by Bobrowski, Kahle, and Skraba shows that for random \v{C}ech and Vietoris--Rips filtrations, the maximal persistence of a $k$-cycle is much smaller, of order $\left(\log n / \log \log n \right)^{1/k}$.