Connectivity threshold for superpositions of Bernoulli random graphs. II

Abstract

Let $G_1,\dots ,G_m$ be independent Bernoulli random subgraphs of the complete graph ${\mathcal{K}}_n$ having variable sizes $X_1,\dots ,X_m\in \{0,1,2,\dots \}$ and densities $Q_1,\dots ,Q_m\in [0,1]$. Letting $n,m\to +\infty$ we establish the connectivity threshold for the union ${\cup }_{i=1}^m{G}_i$ defined on the vertex set of ${\mathcal{K}}_n$. Assuming that $(X_1,Q_1),(X_2,Q_2),\dots ,(X_m,Q_m)$ are independent identically distributed bivariate random variables and $\ln n-\frac{m}{n}E(X_1(1-(1-Q_1{)}^{|X_1-1|})\to c$ we show that $P\{{\cup }_{i=1}^m{G}_i$ is connected$\}\to e^{-e^c}$.The result extends to the case of non-identically distributed random variables $(X_1,Q_1),\dots ,(X_m,Q_m)$ as well.