Daumilas Ardickas | Searcharxiv

arXiv2025

k-connectivity threshold for superpositions of Bernoulli random graphs

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 ${\cal K}_n$.

math.PR↗

arXiv2023

Connectivity threshold for superpositions of Bernoulli random graphs

Let $G_1,\dots, G_m$ be independent Bernoulli random subgraphs of the complete graph ${\cal K}_n$ having variable sizes $x_1,\dots, x_m\in [n]$ and densities $q_1,\dots, q_m\in [0,1]$. Letting $n,m\to+\infty$, we study the connectivity threshold for the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Assuming that the empirical distribution $P_{n,m}$ of the pairs $(x_1,q_1),\dots, (x_m,q_m)$ converges to a probability distribution $P$ we show that the threshold is defined by the mixed moments $κ_n=\iint x(1-(1-q)^{|x-1|})P_{n,m}(dx,dq)$. For $\ln n-\frac{m}{n}κ_n\to-\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 1$ and for $\ln n-\frac{m}{n}κ_n\to+\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 0$. Interestingly, this dichotomy only holds if the mixed moment $\iint x(1-(1-q)^{|x-1|})\ln(1+x)P(dx,dq)<\infty$.

math.PR↗