# Evolution of high-order connected components in random hypergraphs

## TL;DR

This paper studies the emergence and properties of high-order connected components in random hypergraphs, identifying thresholds and behaviors similar to classical graph connectivity but in a hypergraph setting.

## Contribution

It extends classical connectivity results to high-order connectivity in hypergraphs, providing thresholds, component sizes, and hitting time results.

## Key findings

- Determines the asymptotic size of the giant component after its emergence.
- Establishes the threshold for $j$-connectivity in random hypergraphs.
- Proves the hypergraph becomes $j$-connected when the last isolated $j$-set disappears.

## Abstract

We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the evolution of $j$-connected components in the $k$-uniform binomial random hypergraph $\mathcal{H}^k(n,p)$. In particular, we determine the asymptotic size of the giant component shortly after its emergence and establish the threshold at which the $\mathcal{H}^k(n,p)$ becomes $j$-connected with high probability. We also obtain a hitting time result for the related random hypergraph process $\{\mathcal{H}^k(n,M)\}_M$ -- the hypergraph becomes $j$-connected exactly at the moment when the last isolated $j$-set disappears. This generalises well-known results for graphs and vertex-connectivity in hypergraphs.