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On Asymptotic Properties of Bell Polynomials and Concentration of Vertex Degree of Large Random Graphs

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Abstract

We study concentration properties of vertex degrees of n-dimensional Erdős–Rényi 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–Rényi random graphs in the limit \(n,\rho \rightarrow \infty \) such that the ratio \(\rho /\log n \) remains finite or infinitely increases.

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  1. Université de Versailles - Saint-Quentin, 45, Avenue des Etats-Unis, 78035, Versailles, France

    O. Khorunzhiy

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Khorunzhiy, O. On Asymptotic Properties of Bell Polynomials and Concentration of Vertex Degree of Large Random Graphs. J Theor Probab 35, 20–51 (2022). https://doi.org/10.1007/s10959-020-01025-w

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