Connectivity threshold for superpositions of Bernoulli random graphs

Abstract

Let $G_1,\dots ,G_m$ be independent Bernoulli random subgraphs of the complete graph $K_n$ having variable sizes $x_1,\dots ,x_m\in \left[n\right]$ and densities $q_1,\dots ,q_m\in [0,1]$. Letting $n,m\to +\infty$, we study the connectivity threshold for the union ${\cup }_{k=1}^m{G}_k$ defined on the vertex set of $K_n$. 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 $P\{{\cup }_{k=1}^m{G}_k$ is connected$\}\to 1$ and for $\ln n-\frac{m}{n}{\kappa }_n\to +\infty$ we have $P\{{\cup }_{k=1}^m{G}_k$ is connected$\}\to 0$. Interestingly, this dichotomy only holds if the mixed moment $\iint x(1-{(1-q)}^{|x-1|})\ln (1+x)P(dx,dq)<\infty$.