The persistent homology of the Linial-Meshulam process

TL;DR

This paper studies the asymptotic behavior of persistence diagrams in a randomly evolving simplicial complex process, extending known results on Betti numbers of Linial-Meshulam complexes.

Contribution

It provides a detailed analysis of the persistence diagrams' asymptotics for a new class of random complexes, generalizing prior Betti number results.

Findings

Asymptotic behavior of persistence diagrams characterized

Extension of Linial and Peled's Betti number results

Application of local weak convergence and sparse matrix rank results

Abstract

For a fixed dimension $k\ge 1$, let us consider the randomly growing simplical complex on the vertex set $\{1,2,\dots,n\}$ defined as follows: We start with the empty complex, and for each $k+1$-element subset $\sigma$ of $\{1,2,\dots,n\}$, we add $\sigma$ and all of its subsets to the complex at some random time $t_\sigma$, where $(t_\sigma)$ are i.i.d. uniform random elements of $[0,n]$. As the complex evolves, new $k-1$-dimensional cycles are born and then at a later time they die, that is, they get filled in. The notion of persistence diagrams, which is a standard tool in topological data analysis, provides a way to record these birth and death times. In this paper, we understand the asymptotic behavior of the persistence diagrams of the above defined randomly evolving complexes as $n$ goes to infinity. As the single time marginals of the above process are variants of the Linial-Meshulam complex, our results can be viewed as extensions of the results of Linial and Peled on the Betti numbers of the Linial-Meshulam complex. Our proof relies on the notion of local weak convergence of graphs and a generalization of the results of Bordenave, Lelarge and Salez on the rank of sparse random matrices.