Generations of random hypergraphs and random simplicial complexes by the map algebra

TL;DR

This paper introduces a mathematical framework for generating and analyzing random hypergraphs and simplicial complexes using map algebra, providing explicit algorithms and probabilistic descriptions.

Contribution

It develops a graded structure of the map algebra and offers algorithms for generating random hypergraphs and simplicial complexes, linking their distributions to Erdős-Rényi models.

Findings

Explicit probability distributions for associated simplicial complexes and hypergraphs.

Construction of a graded map algebra structure for hypergraph analysis.

Algorithms for generating random hypergraphs and complexes.

Abstract

We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the (lower-)associated independence hypergraph of the random hypergraph in terms of the probability distributions of certain random simplicial complex and certain random independence hypergraph of Erd\"os-R\'enyi type. We construct a graded structure of the map algebra explicitly and give algorithms to generate random hypergraphs and random simplicial complexes.