Maximal persistence in random clique complexes
Abstract
We study the persistent homology of an ErdősRényi 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 Čech and VietorisRips filtrations, the maximal persistence of a $k$cycle is much smaller, of order $\left(\log n / \log \log n \right)^{1/k}$.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.05713
 arXiv:
 arXiv:2209.05713
 Bibcode:
 2022arXiv220905713A
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 Mathematics  Probability
 EPrint:
 14 pages, 2 figures