On rate of convergence to the Poisson law of the number of cycles in the generalized random graphs

TL;DR

This paper establishes a convergence rate of O(1/√n) for the distribution of cycle counts in generalized random graphs with power-law weights to a Poisson distribution, using Chen--Stein methods.

Contribution

It provides a new convergence rate result for cycle counts in weighted random graphs with power-law distributions, extending previous asymptotic analyses.

Findings

Convergence rate of O(1/√n) in total variation distance.

Applicable to graphs with power-law distributed weights.

Method based on Chen--Stein approach and ratio properties.

Abstract

Convergence of order $O(1/\sqrt{n})$ is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is based on the Chen--Stein approach and on the derived properties of the ratio of the sum of squares of random variables and the sum of these variables. These properties can be applied to other asymptotic problems related to generalized random graphs.