Degrees in random $m$-ary hooking networks

TL;DR

This paper investigates the asymptotic degree distribution in random m-ary hooking networks using Pólya urn models, establishing strong laws and Gaussian limit laws with a novel covariance matrix computation method.

Contribution

It introduces a new method for computing the covariance matrix in Pólya urns and applies it to analyze degree structures in m-ary hooking networks, extending understanding of their asymptotic behavior.

Findings

Established strong laws for degree distribution

Derived multivariate Gaussian limit laws

Developed a new method for covariance matrix calculation

Abstract

The theme in this paper is a composition of random graphs and P\'olya urns. The random graphs are generated through a small structure called the seed. Via P\'olya urns, we study the asymptotic degree structure in a random $m$-ary hooking network and identify strong laws. We further upgrade the result to second-order asymptotics in the form of multivariate Gaussian limit laws. We give a few concrete examples and explore some properties with a full representation of the Gaussian limit in each case. The asymptotic covariance matrix associated with the P\'olya urn is obtained by a new method that originated in this paper and is reported in [25].