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

Abstract

AbstractWe 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), withmattachments for each of the 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.