On the strength of connectedness of unions of random graphs
Mindaugas Bloznelis
TL;DR
This paper establishes threshold results for the strength of connectivity in unions of random graphs on n vertices. By introducing the key parameter a (the minimal nonzero degree encountered with positive probability) and the threshold function λ(k) that combines n, m, and the expected non-isolated vertex count, the author shows that the union G_{[n,m]} transitions to being a(k+1)-connected or not ak+1-connected depending on the sign of λ(k) as n grows. The approach combines expansion properties with Poisson convergence for isolated vertices and a detailed blossom/decentralization analysis to derive k-connectivity thresholds, valid in a non-identically distributed setting. The results reveal a stepwise growth of connectivity strength in multiples of a and provide a flexible framework for unions of diverse random subgraphs, with explicit lemmas bounding the probability of small disconnecting structures and deriving degree-based concentration. The findings offer principled insight into how the overlap and density of contributing subgraphs shape global connectivity, informing models of networks with overlapping communities and random clique constituents.
Abstract
Let $G_1,\dots, G_m$ be independent identically distributed random subgraphs of the complete graph ${\cal K}_n$. We analyse the threshold behaviour of the strength of connectedness of the union $\cup_{i=1}^mG_i$ defined on the vertex set of ${\cal K}_n$. Let $a=\min\{t\ge 1:\, \PP\{δ(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $λ(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} \E X$, where $X$ stands for the number of non isolated vertices of $G_1$. Letting $n,m\to+\infty$ we show that $\PP\{\cup_{i=1}^mG_i$ is $a(k+1)$-connected$\} \to 1 $ for $λ(k)\to -\infty$, and $\PP\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $λ(k)\to +\infty$. In particular, the connectivity strength of the union graph $\cup_{i=1}^mG_i$ increases in steps of size $a$. Our results are obtained in a more general setting where the contributing random subgraphs do not need to be identically distributed.