Subcritical random hypergraphs, high-order components, and hypertrees

Oliver Josef Nikolaus Cooley, Wenjie Fang, Nicola Del Giudice, Mihyun Kang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish
Title of host publication16th Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2019
EditorsJ. Ian Munro, Marni Michna
Pages111-118
Number of pages8
ISBN (Electronic)9781510879942
Publication statusPublished - 2019
EventAnalytic Algorithmics and Combinatorics - Westin San Diego, San Diego, United States
Duration: 6 Jan 20197 Jan 2019

Conference

ConferenceAnalytic Algorithmics and Combinatorics
Abbreviated titleANALCO19
CountryUnited States
CitySan Diego
Period6/01/197/01/19

ASJC Scopus subject areas

  • Applied Mathematics
  • Materials Chemistry
  • Discrete Mathematics and Combinatorics

Fields of Expertise

  • Information, Communication & Computing

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