Essential edges in Poisson random hypergraphs

TL;DR

This paper investigates the asymptotic proportion of essential hyperedges in Poisson random hypergraphs and explores their relation to the 2-core in random graphs, extending previous work on identifiable vertices.

Contribution

It establishes the limiting proportion of essential hyperedges in Poisson hypergraphs and discusses their connection to the 2-core in random graphs, providing new insights into hypergraph structure.

Findings

Limiting proportion of essential hyperedges determined

Relation between essential edges and the 2-core analyzed

Extension of previous results on identifiable vertices

Abstract

Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process "identifiable". Also any hyperedge all of whose vertices are removed is called "identifiable". We say that a hyperedge is "essential" if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal sub-graph with minimal vertex degree 2.