On asymptotic behavior of Bell polynomials and concentration of vertex degree of large random graphs

Understand

We study concentration properties of vertex degrees of $n$-dimensional Erdos-R\'enyi random graphs with the 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 ${\cal B}_k(\rho), k\in {\bf N}$ that represent moments of the Poisson probability distribution ${\cal P}(\rho)$.
• 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 the upper bounds for the deviation probabilities of the normalized maximal vertex degree of the Erdos-R\'enyi random graphs in the limit $n,\rho\to\infty$ such that the ratio $\rho/\log n $ remains finite or infinitely increases.