Random Uniform and Pure Random Simplicial Complexes

TL;DR

This paper introduces methods to analyze properties of random uniform and pure simplicial complexes, providing bounds for topological and combinatorial features, and explores the implications for monotone boolean functions, supporting the Evasiveness conjecture.

Contribution

It presents new probabilistic models for uniform and pure simplicial complexes and connects these models to the behavior of monotone boolean functions, advancing understanding of their typical properties.

Findings

Bounds for topological and combinatorial properties of random complexes

Most monotone boolean functions are evasive

Supports the Evasiveness conjecture for non-symmetric functions

Abstract

In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties of a complex under this measure. We are able to determine or present bounds for a number of topological and combinatorial properties. We also study the random pure simplicial complex of dimension $d$, generated by letting any subset of size $d+1$ of a set of $n$ vertices be a facet with probability $p$ and considering the simplicial complex generated by these facets. We compare the behaviour of these models for suitable values of $d$ and $p$. Finally we use the equivalence between simplicial complexes and monotone boolean functions to study the behaviour of typical such functions. Specifically we prove that most monotone boolean functions are evasive, hence proving that the well known Evasiveness conjecture is generically true for monotone boolean functions without symmetry assumptions.