k-connectivity threshold for superpositions of Bernoulli random graphs

TL;DR

This work identifies the $k$-connectivity threshold for the union of $m$ independent Bernoulli random subgraphs of the complete graph on $n$ vertices, where the subgraphs have randomly sized vertex sets and edge densities. The authors derive a precise threshold function $\lambda_{n,m,k}=\ln n+(k-1)\ln(m/n)-(m/n)\kappa$ (with $\kappa=\mathbb{E}[\tilde{X}h(\tilde{X},Q)]$) that governs the transition, under the regime $m=\Theta(n\ln n)$ and finite moments like $\tau^*>0$, $\eta_2<\infty$, and $\mu'<\infty$. A two-lemma strategy is employed: (i) a degree-based condition showing a vertex of degree $k-1$ triggers disconnection, and (ii) a component-avoidance argument ensuring no small components persist, which together yield $\mathbb{P}\{G_{[n,m]} \text{ is vertex } k\text{-connected}\}$ converging to 1 in the right regime and to 0 otherwise. These results extend connectivity thresholds to superpositions of random graphs with random vertex counts, offering a precise asymptotic picture for network reliability in such models.

Abstract

Let $G_1,\dots, G_m$ be independent identically distributed Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having vertex sets of random sizes $X_1,\dots, X_m\in \{0,1,2,\dots\}$ and random edge densities $Q_1,\dots, Q_m\in [0,1]$. Assuming that each $G_i$ has a vertex of degree $1$ with positive probability, we establish the $k$-connectivity threshold as $n,m\to+\infty$ for the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2
• proof : Proof of Theorem \ref{['Theorem_1']}
• Lemma 3
• proof : Proof of Lemma \ref{['sr']}
• Lemma 4
• proof
• proof : Proof of Lemma \ref{['Lemma_1']}
• Lemma 5
• Lemma 6