# On the growth of a superlinear preferential attachment scheme

## TL;DR

This paper rigorously analyzes the degree distribution in a superlinear preferential attachment model, confirming the condensation phenomenon where one node dominates with infinite degree, and provides law of large numbers and fluctuation results.

## Contribution

It establishes rigorous law of large numbers and fluctuation results for degree counts in superlinear preferential attachment graphs, extending previous physical models.

## Key findings

- Confirmation of condensation with a single dominant node
- Law of large numbers for degree counts
- Fluctuation results around the degree distribution

## Abstract

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as $n\uparrow\infty$, for the counts of the number of nodes with degree $k\geq 1$ at time $n\geq 1$. These limits rigorously verify and extend a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.