Unimodality of the Expectation of Betti Numbers for Bernoulli Random Quota 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 $m$.