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

TL;DR

This paper investigates the asymptotic behavior of Bell polynomials and their relation to vertex degree concentration in large Erdős-Rényi graphs, providing bounds on deviation probabilities as graph size and density grow.

Contribution

It establishes the asymptotic connection between Bell polynomials and moments of vertex degrees, deriving bounds on degree deviation probabilities in large random graphs.

Findings

Bell polynomials approximate moments of vertex degrees

Asymptotic behavior of Bell polynomials analyzed for large parameters

Derived upper bounds for maximum degree deviations

Abstract

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.