# Preferential Attachment Random Graphs with Edge-Step Functions

## TL;DR

This paper introduces a new preferential attachment random graph model with edge-step functions, analyzing how these functions influence the degree distribution, especially under regular variation conditions, revealing power-law behaviors and degree concentration properties.

## Contribution

It establishes the relationship between edge-step functions' regular variation and the resulting degree distribution in preferential attachment graphs, extending understanding of their asymptotic properties.

## Key findings

- Power-law degree distribution when the edge-step function is regularly varying with index > -1.
- Vertices with small degrees become negligible when the index ≤ -1.
- The degree distribution's exponent relates to the regular variation index of the edge-step function.

## Abstract

We propose a random graph model with preferential attachment rule and \emph{edge-step functions} that govern the growth rate of the vertex set. We study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function $f$ is a \emph{monotone regularly varying function} at infinity, the sequence of graphs associated to it obeys a power-law degree distribution whose exponent is related to the index of regular variation of $f$ at infinity whenever said index is greater than $-1$. When the regularly variation index is less than or equal to $-1$, we show that the proportion of vertices with degree smaller than any given constant goes to $0$ a. s..