Maximal persistence in random clique complexes

TL;DR

This paper investigates the maximal persistence of cycles in Erdős–Rényi random clique complexes, revealing that the persistence scales polynomially with the number of vertices, contrasting with geometric models where it grows logarithmically.

Contribution

It provides the first rigorous analysis of the order of maximal persistence in random clique complexes, establishing a polynomial growth rate and contrasting it with geometric models.

Findings

Maximal persistence of a k-cycle is roughly n^{1/k(k+1)}

Persistence scales polynomially with number of vertices n

Contrasts with geometric models where persistence is logarithmic

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 $\left(\log n / \log \log n \right)^{1/k}$.