关于随机图并集连通性的强度
On the strength of connectedness of unions of random graphs
AI总结：
研究随机图并集的连通性强度阈值行为，证明其在特定条件下随着参数变化而逐步增加。
摘要：
令G₁,…,Gₘ是完全图Kₙ的独立同分布随机子图。我们分析并集∪_{i=1}^mG_i的连通性强度的阈值行为，该并集在Kₙ的顶点集上定义。令a=min{t≥1:P{δ(G₁)=t>0}}为最小非零顶点度数，以正概率出现。给定k≥0，令λ(k)=ln n +k ln(m/n) - (m/n)E X，其中X表示G₁中非孤立顶点的数量。当n,m→+∞时，我们证明当λ(k)→-∞时，P{∪_{i=1}^mG_i是a(k+1)-连通}→1，当λ(k)→+∞时，P{∪_{i=1}^mG_i是ak+1-连通}→0。特别地，并集图∪_{i=1}^mG_i的连通性强度以步长a增加。我们的结果是在更一般的设定下获得的，其中贡献的随机子图不需要同分布。
英文摘要：
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 $a=\min\{t\ge 1:\, {\bf P}\{δ(G_1)=t>0\}\}$ be the minimal non zero vertex degree attained with positive probability. Given $k\ge 0$ let $λ(k)=\ln n+k\ln\frac{m}{n}-\frac{m}{n} {\bf E} X$, 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 $λ(k)\to -\infty$, and ${\bf P}\{\cup_{i=1}^mG_i$ 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.
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