Random Simplicial Complexes

TL;DR

This paper introduces a multi-parameter random simplicial complex model that generalizes existing models, revealing that topological properties depend on parameter combinations and have convex thresholds, advancing understanding of complex topological structures.

Contribution

It presents a new multi-parameter model encompassing known models, analyzing how topological properties depend on parameter combinations and convex thresholds.

Findings

Topological properties depend on the entire set of probability parameters.

Thresholds for properties are convex sets, not single values.

The model includes Linial-Meshulam and clique complexes as special cases.

Abstract

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.