Degree sequences of sufficiently dense random uniform hypergraphs

TL;DR

This paper derives an asymptotic enumeration formula for simple r-uniform hypergraphs with specified degree sequences, especially near regular sequences, and compares their degree distributions to probabilistic models.

Contribution

It provides the first explicit asymptotic enumeration formula for such hypergraphs and analyzes the properties of their degree sequences under various conditions.

Findings

Derived an asymptotic enumeration formula for hypergraphs with given degree sequences.

Established conditions for the existence of solutions to the system of equations.

Compared degree sequences of random hypergraphs to binomial and hypergeometric models.

Abstract

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$-uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum.