Unimodality of the Expectation of Betti Numbers for Bernoulli Random Quota Complexes

TL;DR

This paper investigates the expected Betti numbers of Bernoulli-weighted random quota complexes, demonstrating that these expectations are unimodal across different homology dimensions, revealing a structured pattern in their topological complexity.

Contribution

It introduces the study of Betti number expectations in Bernoulli random quota complexes and proves their unimodality, a novel insight into their topological behavior.

Findings

Expected Betti numbers are unimodal in the homology dimension.

The study provides a new understanding of the topological structure of Bernoulli quota complexes.

The results suggest predictable patterns in the complexity of random simplicial complexes.

Abstract

We study certain random simplicial complexes, called random quota complexes. A quota complex on $N+1$ weighted vertices is constructed by adding an $n$-simplex if the sum of the weights of the vertices is below a given quota, $q$. In this paper, the weights of the vertices are chosen i.i.d. with a Bernoulli distribution. The main result of this paper is that the expectation of the $m^{\textrm{th}}$ Betti number, i.e., the dimension of the $m^{\textrm{th}}$ homology group, is unimodal in $m$.