Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The persistent homology of the Linial-Meshulam process

András Mészáros

TL;DR

This work analyzes the persistent homology of a random high dimensional Linial-Meshulam type filtration on [n], deriving precise limiting behavior for persistent Betti numbers and persistence diagrams as n grows. By combining local weak convergence of graphs with rank analysis of sparse random matrices, the authors obtain deterministic limits hat beta for beta^{r,s}_{k-1} and limiting verbose and standard persistence diagrams hat xi Ver and hat xi. The limits are characterized via fixed point equations and associated functions Lambda, lambda, and fixed point structures, revealing phase transition phenomena that extend Linial and Peled results to persistence data. The results establish a rigorous link between random simplicial filtrations and sparse matrix ranks, enabling explicit computation of limiting topological observables and providing tools for analyzing random high dimensional data shapes. Overall, the paper advances the theoretical understanding of random filtrations by delivering explicit limit objects for persistent homology and connecting topological data analysis with random matrix theory and branching processes.

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 $σ$ of $\{1,2,\dots,n\}$, we add $σ$ and all of its subsets to the complex at some random time $t_σ$, where $(t_σ)$ 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.

The persistent homology of the Linial-Meshulam process

TL;DR

This work analyzes the persistent homology of a random high dimensional Linial-Meshulam type filtration on [n], deriving precise limiting behavior for persistent Betti numbers and persistence diagrams as n grows. By combining local weak convergence of graphs with rank analysis of sparse random matrices, the authors obtain deterministic limits hat beta for beta^{r,s}_{k-1} and limiting verbose and standard persistence diagrams hat xi Ver and hat xi. The limits are characterized via fixed point equations and associated functions Lambda, lambda, and fixed point structures, revealing phase transition phenomena that extend Linial and Peled results to persistence data. The results establish a rigorous link between random simplicial filtrations and sparse matrix ranks, enabling explicit computation of limiting topological observables and providing tools for analyzing random high dimensional data shapes. Overall, the paper advances the theoretical understanding of random filtrations by delivering explicit limit objects for persistent homology and connecting topological data analysis with random matrix theory and branching processes.

Abstract

For a fixed dimension , let us consider the randomly growing simplical complex on the vertex set defined as follows: We start with the empty complex, and for each -element subset of , we add and all of its subsets to the complex at some random time , where are i.i.d. uniform random elements of . As the complex evolves, new -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 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.

Paper Structure

This paper contains 24 sections, 38 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Persistence diagrams
  3. Topological data analysis -- A motivation for studying persistent homology
  4. Recovering the results of Linial and Peled
  5. Preliminaries
  6. Local weak convergence of graphs
  7. Multi-type Galton-Watson trees
  8. The convergence of persistent Betti numbers
  9. A theorem on the rank of sparse of matrices
  10. The convergence of persistent Betti numbers
  11. The local weak limit of matrices coming from the Linial-Meshulam filtration -- The proof of Lemma \ref{['lemmaMnlimit']}
  12. The fixed points of the function $\exp(-c(1-qt)^k)$ -- The proof of Lemma \ref{['lemmaFixedpoints']}
  13. Convergence of persistence diagrams
  14. Convergence of the verbose persistence diagrams
  15. Convergence of the persistence diagrams
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $r,s\in [0,\infty)$. As $n$ tends to infinity, the random variables converge in probability to some constant $\hat{\beta}_{k-1}^{r,s}$ defined as

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Theorem 1.5: Linial and Peled linial2016phase
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.1
  • Lemma 3.2
  • ...and 51 more