# Giant descendant trees, matchings and independent sets in the age-biased attachment graphs

## TL;DR

This paper analyzes age-biased attachment graph models, revealing the limiting behavior of descendant trees and evaluating greedy algorithms for matchings and independent sets, with results on convergence and distribution.

## Contribution

It introduces a detailed analysis of descendant trees and greedy algorithms in age-biased attachment graphs, providing new insights into their asymptotic properties.

## Key findings

- Scaled descendant tree size converges to a mixture of beta-distributions for m=1.
- For m>1, the descendant tree size converges to 1 almost surely.
- Greedy algorithms achieve specific limiting fractions of vertices in matchings and independent sets.

## Abstract

We study two models of an age-biased graph process: the $\delta$-version of the preferential attachment graph model (PAM) and the uniform attachment graph model (UAM), with $m$ attachments for each of incoming vertices. We show that almost surely the scaled size of a breadth-first (descendant) tree rooted at a fixed vertex converges, for $m=1$, to a limit whose distribution is a mixture of two beta-distributions and a single beta-distribution respectively, and that for $m>1$ the limit is $1$. We also analyze the likely performance of two greedy (online) algorithms, for a large matching set and a large independent set, and determine--for each model and each greedy algorithm--both a limiting fraction of vertices involved and an almost sure convergence rate.