On the strength of connectedness of unions of random graphs

TL;DR

This paper investigates the threshold behavior of the connectedness strength of unions of independent random subgraphs of a complete graph, revealing stepwise increases in connectivity as parameters vary.

Contribution

It extends previous models by analyzing unions of non-identically distributed random subgraphs and characterizes the sharp thresholds for their connectivity levels.

Findings

Connectivity increases in steps of size 'a' as parameters change.

Thresholds for connectivity are characterized by the parameter λ(k).

Results hold for non-identically distributed subgraphs.

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 $a=\min\{t\ge 1:\, {\bf P}\{\delta(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $\lambda(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, where $X$ stands for the number of non isolated vertices of $G_1$. Letting $n,m\to+\infty$ we show that ${\bf P}\{\cup_{i=1}^mG_i$ is $a(k+1)$-connected$\} \to 1 $ for $\lambda(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ is $ak+1$-connected$\} \to 0 $ for $\lambda(k)\to +\infty$. 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.