The Fine Structure of Preferential Attachment Graphs I: Somewhere-Denseness

TL;DR

This paper demonstrates that preferential attachment graphs are asymptotically almost surely somewhere-dense by showing they contain large subdivided cliques, challenging the applicability of sparse graph algorithms.

Contribution

It introduces improved concentration bounds for vertex degrees and proves that preferential attachment graphs contain large subdivided cliques, establishing their somewhere-denseness.

Findings

Preferential attachment graphs contain large subdivided cliques of size at least ()^{1/4}.

They are asymptotically almost surely somewhere-dense.

Concentration bounds for vertex degrees are sharply concentrated for large degrees.

Abstract

Preferential attachment graphs are random graphs designed to mimic properties of typical real world networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We use improved concentration bounds for vertex degrees to show that preferential attachment graphs contain asymptotically almost surely (a.a.s.) a one-subdivided clique of size at least $(\log n)^{1/4}$. Therefore, preferential attachment graphs are a.a.s somewhere-dense. This implies that algorithmic techniques developed for sparse graphs are not directly applicable to them. The concentration bounds state: Assuming that the exact degree $d$ of a fixed vertex (or set of vertices) at some early time $t$ of the random process is known, the probability distribution of $d$ is sharply concentrated as the random process evolves if and only if $d$ is large at time $t$.