A binomial random multigraph

TL;DR

The paper introduces the binomial random multigraph model $\mathbb{G}(n,p;\mathcal{H})$ built from a base hypergraph $\mathcal{H}$, where for each hyperedge a random pair is chosen and its edge is added with probability $p$. It establishes a mixture decomposition over 2-uniform shadows, analyzes degree distributions and edge counts as mixtures of Binomial and Hypergeometric variables, and proves monotonicity in both $p$ and $\mathcal{H}$. A central result is a gap-threshold phenomenon: for properties with a threshold in $\mathbb{G}(n,p)$, there exist distinct scale sequences $b_n^{\ast}$ and $c_n^{\ast}$ such that the property remains unlikely below $b_n^{\ast}$ but becomes likely above $c_n^{\ast}$ in the multigraph setting, illustrating how hypergraph structure affects thresholds. The work also provides concrete threshold results for simplicity and connectivity in the $k=3$ setting, and analyzes triangle counts in several hypergraph-driven variants, highlighting both analogies and new dependencies relative to classical $\mathbb{G}(n,p)$ theory.

Abstract

Fix a positive integer $n$, a real number $p\in (0,1]$, and a (perhaps random) hypergraph $\mathcal{H}$ on $[n]$. We introduce and investigate the following random multigraph model, which we denote $\mathbb{G}(n,p\, ; \,\mathcal{H})$: begin with an empty graph on $n$ vertices, which are labelled by the set $[n]$. For every $H\in \mathcal{H}$ choose, independently from previous choices, a doubleton from $H$, say $D = \{i,j\} \subset H$, uniformly at random and then introduce an edge between the vertices $i$ and $j$ in the graph with probability $p$, where each edge is introduced independently of all other edges.

Theorems & Definitions (41)

• Definition 1: Threshold functions
• Definition 2: Threshold functions in $\mathbb{G}(n,p\, ; \,\mathbb{H}_{n,q;k})$
• Conjecture 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7