## Abstract

One of the central topics in the theory of random graphs deals with the phase transition in the order of the largest components. In the binomial random graph G(n, p), the threshold for the appearance of the unique largest component (also known as the giant component) is p
_{g} = n
^{−}
^{1} . More precisely, when p changes from (1 − ε)p
_{g} (subcritical case) to p
_{g} and then to (1 + ε)p
_{g} (supercritical case) for ε > 0, with high probability the order of the largest component increases smoothly from O(ε
^{−}
^{2} log(ε
^{3} n)) to Θ(n
^{2} /
^{3} ) and then to (1 ± o(1))2εn. Furthermore, in the supercritical case, with high probability the largest components except the giant component are trees of order O(ε
^{−}
^{2} log(ε
^{3} n)), exhibiting a structural symmetry between the subcritical random graph and the graph obtained from the supercritical random graph by deleting its giant component. As a natural generalisation of random graphs and connectedness, we consider the binomial random kuniform hypergraph H
^{k} (n, p) (where each k-tuple of vertices is present as a hyperedge with probability p independently) and the following notion of high-order connectedness. Given an integer 1 ≤ j ≤ k − 1, two sets of j vertices are called j-connected if there is a walk of hyperedges between them such that any two consecutive hyperedges intersect in at least j vertices. A j-connected component is a maximal collection of pairwise j-connected j-tuples of vertices. Recently, the threshold for the appearance of the giant j-connected component in H
^{k} (n, p) and its order were determined. In this article, we take a closer look at the subcritical random hypergraph. We determine the structure and size (i.e. number of hyperedges) of the largest j-connected components, with the help of a certain class of “hypertrees” and related objects. In our proofs, we combine various probabilistic and enumerative techniques, such as generating functions and couplings with branching processes. Our study will pave the way to establishing a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical random hypergraph by deleting its giant j-connected component.

Original language | English |
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Title of host publication | 16th Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2019 |

Editors | J. Ian Munro, Marni Michna |

Pages | 111-118 |

Number of pages | 8 |

ISBN (Electronic) | 9781510879942 |

Publication status | Published - 2019 |

Event | Analytic Algorithmics and Combinatorics - Westin San Diego, San Diego, United States Duration: 6 Jan 2019 → 7 Jan 2019 |

### Conference

Conference | Analytic Algorithmics and Combinatorics |
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Abbreviated title | ANALCO19 |

Country/Territory | United States |

City | San Diego |

Period | 6/01/19 → 7/01/19 |

## ASJC Scopus subject areas

- Applied Mathematics
- Materials Chemistry
- Discrete Mathematics and Combinatorics

## Fields of Expertise

- Information, Communication & Computing