# On the dense Preferential Attachment Graph models and their graphon induced counterpart

## TL;DR

This paper compares the dense Preferential Attachment Graph (PAG) model with its graphon-based W-random graph counterpart, providing bounds on their expected distance and insights into their convergence behavior.

## Contribution

It introduces a coupling method to bound the expected jumble norm distance between PAG and W-random graphs, advancing understanding of their relationship.

## Key findings

- Expected jumble norm distance bounded by O(log^2 n * n^{-1/3})
- Universal lower bound established independent of coupling
- Analysis enhances understanding of PAG convergence to graphons

## Abstract

Letting $\mathcal{M}$ denote the space of finite measures on $\mathbb{N}$, and $\mu_\lambda\in\mathcal{M}$ denote the Poisson distribution with parameter $\lambda$, the function $W:[0,1]^2\to\mathcal{M}$ given by \[ W(x,y)=\mu_{c\log x\log y} \] is called the PAG graphon with density $c$. It is known that this is the limit, in the multigraph homomorphism sense, of the dense Preferential Attachment Graph (PAG) model with edge density $c$. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way. The aim of this paper is to compare the dense PAG model with the W-random graph model obtained from the corresponding graphon. Motivated by the multigraph limit theory, we investigate the expected jumble norm distance of the two models in terms on the number of vertices $n$. We present a coupling for which the expectation can be bounded from above by $O(\log^2 n\cdot n^{-1/3})$, and provide a universal lower bound that is coupling independent, but with a worse exponent.