Modularity of preferential attachment graphs

TL;DR

This paper analyzes the community structure of preferential attachment graphs, proving that their modularity diminishes as the number of connections per new vertex increases, and introduces new concentration results for graph parameters.

Contribution

It proves that the modularity of preferential attachment graphs tends to zero as the number of edges per new vertex grows, resolving a 2016 conjecture and extending previous concentration results.

Findings

Modularity of G_n^h is with high probability bounded above by a function tending to 0 as h increases.

Introduces a new measure μ for vertex subsets, proportional to their average volume, for analyzing graph properties.

Provides novel concentration results for volume and edge density parameters of vertex subsets.

Abstract

We study the preferential attachment model $G_n^h$. A graph $G_n^h$ is generated from a finite initial graph by adding new vertices one at a time. Each new vertex connects to $h\ge 1$ already existing vertices, and these are chosen with probability proportional to their current degrees. We are particularly interested in the community structure of $G_n^h$, which is expressed in terms of the so-called modularity. We prove that the modularity of $G_n^h$ is with high probability upper bounded by a function that tends to $0$ as $h$ tends to infinity. This resolves the conjecture of Prokhorenkova, Pralat, and Raigorodskii from 2016. As a byproduct, we obtain novel concentration results (which are interesting in their own right) for the volume and edge density parameters of vertex subsets of $G_n^h$. The key ingredient here is the definition of the function $\mu$, which serves as a natural measure for vertex subsets, and is proportional to the average size of their volumes. This extends previous results on the topic by Frieze, Pralat, P\'erez-Gim\'enez, and Reiniger from 2019.