# Preferential attachment without vertex growth: emergence of the giant component

## TL;DR

This paper analyzes a variant of the preferential attachment process without vertex addition, showing how a giant component emerges in a fixed vertex set by adding edges proportionally to endpoint degrees, confirming a prior conjecture.

## Contribution

It establishes the asymptotic size of the giant component in a degree-proportional edge addition process, extending understanding of phase transitions in fixed-vertex random graphs.

## Key findings

- Confirmed the conjecture of Pittel (2010) on giant component emergence.
- Determined the asymptotic size of the giant component in the supercritical phase.
- Introduced a simple conditioning method using the configuration model.

## Abstract

We study the following preferential attachment variant of the classical Erdos-Renyi random graph process. Starting with an empty graph on n vertices, new edges are added one-by-one, and each time an edge is chosen with probability roughly proportional to the product of the current degrees of its endpoints (note that the vertex set is fixed). We determine the asymptotic size of the giant component in the supercritical phase, confirming a conjecture of Pittel from 2010. Our proof uses a simple method: we condition on the vertex degrees (of a multigraph variant), and use known results for the configuration model.