Creation and Growth of Components in a Random Hypergraph Process

TL;DR

This paper analyzes the formation and growth of specific connected components in a random hypergraph process, revealing their expected counts and sizes during the phase transition, especially as parameters grow large.

Contribution

It provides new asymptotic results on the expected number and size of ll-components in a random hypergraph process, extending understanding of phase transitions.

Findings

Expected number of ll-components tends to 1 as parameters grow.

Size of the largest ll-component is approximately proportional to ^{1/3} ll^{1/3} n^{2/3}.

Results give insight into the structure of giant components during phase transition.

Abstract

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.