  Let $G_1,\dots, G_m$ be independent Bernoulli random subgraphs of the
complete graph ${\cal K}_n$ having variable sizes $X_1,\dots, X_m\in
\{0,1,2,\dots\}$ and densities $Q_1,\dots, Q_m\in [0,1]$. Letting
$n,m\to+\infty$ we establish the connectivity threshold for the union
$\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Assuming that
$(X_1,Q_1), (X_2,Q_2),\dots, (X_m,Q_m)$ are independent identically distributed
bivariate random variables and $\ln n
-\frac{m}{n}E\bigl(X_1(1-(1-Q_1)^{|X_1-1|}\bigr)\to c$ we show that
$P\{\cup_{i=1}^mG_i$ 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.
