The Shadow knows: Empirical Distributions of Minimum Spanning Acycles and Persistence Diagrams of Random Complexes

TL;DR

This paper studies the distribution of weights in random minimum spanning acycles and their persistence diagrams, extending classical results to higher-dimensional complexes and introducing the concept of the shadow for empirical distribution convergence.

Contribution

It introduces the empirical distribution convergence of weights in random complexes to a measure based on the shadow, generalizing known results to higher dimensions and persistence diagrams.

Findings

Empirical distributions of weights converge to a measure based on the shadow.

The approach generalizes Frieze's MST result to higher-dimensional complexes.

Provides insights into the distribution of death times in persistence diagrams.

Abstract

In 1985, Frieze showed that the expected sum of the edge weights of the minimum spanning tree (MST) in the uniformly weighted graph converges to $\zeta(3)$. Recently, Hino and Kanazawa extended this result to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analog -- the Minimum Spanning Acycle (MSA). Our work goes beyond and describes the histogram of all the weights in this random MST and random MSA. Specifically, we show that their empirical distributions converge to a measure based on a concept called the shadow. The shadow of a graph is the set of all the missing transitive edges, and, for a simplicial complex, it is a related topological generalization. As a corollary, we obtain a similar claim for the death times in the persistence diagram corresponding to the above-weighted complex, a result of interest in applied topology.