Connectivity threshold for superpositions of Bernoulli random graphs. II (2311.09317v1)

Abstract: 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.