2022/09/13 by Ayat Ababneh, Ababneh, Ayat, Matthew Kahle +1 · 1 citation
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2209.05713
openalex publication_date 2022/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the persistent homology of an Erdős--Rényi random clique complex filtration on n vertices. Here, each edge e appears at a time pe ∈ [0,1] chosen uniform randomly in the interval, and the persistence of a cycle σ is defined as p2 / p1, where p1 and p2 are the birth and death times of the cycle respectively. We show that for fixed k ≥ 1, with high probability the maximal persistence of a k-cycle is of order roughly n1/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 Vietoris--Rips filtrations, the maximal persistence of a k-cycle is much smaller, of order (log n / log log n )1/k.