# Diameter of P.A. random graphs with edge-step functions

## TL;DR

This paper establishes bounds for the diameter of preferential attachment random graphs driven by a function, showing it scales as the inverse of the regular variation index for certain functions.

## Contribution

It provides sharp bounds and characterizations for the diameter of these graphs based on the regular variation properties of the attachment function.

## Key findings

- Diameter scales as 1/γ for regularly varying functions with index -γ.
- Results are sharp for slowly varying functions.
- Diameter grows polynomially with the number of vertices.

## Abstract

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when $f$ is a \textit{regularly varying function at infinity} with strictly negative index of regular variation~$-\gamma$. For this particular class, we prove a characterization for the diameter that depends only on~$-\gamma$. More specifically, we prove that the diameter of such graphs is of order $1/\gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of \textit{slowly varying functions} are also obtained.