Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

TL;DR

This paper extends Frieze's limit theorem to higher dimensions by analyzing the minimum spanning acycle in the Linial-Meshulam process, linking it to persistent homology and showing its expected weight scales as O(n^{d-1}).

Contribution

It introduces the concept of spanning acycles as a higher dimensional analogue of spanning trees and establishes their expected weight behavior in the Linial-Meshulam process.

Findings

Expected weight of minimum spanning acycle is O(n^{d-1})

Connects spanning acycles to persistent homology

Generalizes Frieze's theorem to higher dimensions

Abstract

This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$.