We study concentration properties of vertex degrees of n-dimensional Erdős–Renyi random graphs with edge probability $$\rho /n$$ by means of high moments of these random variables in the limit when n and $$\rho $$ tend to infinity. These moments are asymptotically close to one-variable Bell polynomials $${{\mathcal {B}}}_k(\rho ), k\in {{\mathbb {N}}}$$ , that represent moments of the Poisson probability distribution $${{\mathcal {P}}}(\rho )$$ . We study asymptotic behavior of the Bell polynomials and modified Bell polynomials for large values of k and $$\rho $$ with the help of the local limit theorem for auxiliary random variables. Using the results obtained, we get upper bounds for the deviation probabilities of the normalized maximal vertex degree of the Erdős–Renyi random graphs in the limit $$n,\rho \rightarrow \infty $$ such that the ratio $$\rho /\log n $$ remains finite or infinitely increases.

中文翻译：

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贝尔多项式的渐近性质与大随机图顶点度的集中

我们通过当n和$$\rho $$趋向于极限时这些随机变量的高矩来研究边概率为$$\rho /n$$的n维Erdős-Renyi随机图的顶点度的集中特性无限。这些矩渐近接近于单变量贝尔多项式 $${{\mathcal {B}}}_k(\rho ), k\in {{\mathbb {N}}}$$ ，代表泊松概率的矩分布 $${{\mathcal {P}}}(\rho )$$ 。我们借助辅助随机变量的局部极限定理研究了贝尔多项式和修正贝尔多项式在 k​​ 和 $$\rho $$ 的大值下的渐近行为。使用获得的结果，我们得到极限 $$n 内 Erdős-Renyi 随机图的归一化最大顶点度的偏差概率的上限，