Connectivity threshold for superpositions of Bernoulli random graphs

Abstract

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 $\kappa_n=\iint x(1-(1-q)^{|x-1|})P_{n,m}(dx,dq)$. For $\ln n-\frac{m}{n}\kappa_n\to-\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 1$ and for $\ln n-\frac{m}{n}\kappa_n\to+\infty$ we have $P\{\cup_{i=1}^mG_i$ is connected$\}\to 0$. Interestingly, this dichotomy only holds if the mixed moment $\iint x(1-(1-q)^{|x-1|})\ln(1+x)P(dx,dq)<\infty$.