k-connectivity threshold for superpositions of Bernoulli random graphs

Abstract: Let $G_1,\dots, G_m$ be independent identically distributed Bernoulli random subgraphs of the complete graph ${\cal K}n$ having vertex sets of random sizes $X_1,\dots, X_m\in {0,1,2,\dots}$ and random edge densities $Q_1,\dots, Q_m\in [0,1]$. Assuming that each $G_i$ has a vertex of degree $1$ with positive probability, we establish the $k$-connectivity threshold as $n,m\to+\infty$ for the union $\cup{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$.