Connectivity threshold for superpositions of Bernoulli random graphs. II

Let \(G_1\), ..., \(G_m\) be independent Bernoulli random subgraphs of the complete graph \(\mathcal{K}_n\) having random sizes \(X_1,\dots, X_m\in \{0,1,2,\dots\}\) and edge densities \(Q_1\), ..., \(Q_m\in [0,1]\). Letting \(n,m\to+\infty\) we establish the connectivity threshold for the union \( \bigcup_{i=1}^mG_i\) defined on the vertex set of \(\mathcal{K}_n\). We show that

$$ \textbf{P} \bigl \{ \bigcup_{i=1}^m G_i \ \hbox{is connected}\ \bigr \}= e^{-e^{\lambda^*_{n,m}}}+o(1) , $$

where \(\lambda^{*}_{n,m}= \ln n - \frac{1}{n} \sum\nolimits_{i=1}^{m} \textbf{E} X_{i}(1-(1-Q_i)^{|X_i-1|})\).