Modularity of preferential attachment graphs

TL;DR

This work analyzes the modularity of preferential attachment graphs $G_n^h$ and proves that whp mod$(G_n^h)$ vanishes as $h\to\infty$ by establishing sharp concentration results for volumes and edge counts of vertex subsets. The authors introduce a natural measure $\mu$ on mini-vertex sets in the associated random tree $T_{hn}$ and develop martingale-based concentration techniques (Azuma-Hoeffding and Freedman) to bound $\mathrm{vol}(S)$, $e(S)$, and $e(S,V\setminus S)$ for all $S\subseteq V$, linking them to $\mu(\tilde{S})$. The main result provides an explicit upper bound $\mathrm{mod}(G_n^h) \le (1+\varepsilon) f(h)/\sqrt{h}$ with $f(h) \sim 3\sqrt{2\ln 2}\sqrt{\ln h}$, resolving a 2016 conjecture by Prokhorenkova, Prałat and Raigorodskii and showing that standard PA graphs have diminishing modularity for large $h$. The paper additionally develops a two-phase construction via $T_{hn}$ that connects PA dynamics to a related random graph $\hat{G}$ with edge prob $1/(2\sqrt{ij})$, enabling the asymptotic analysis. Overall, the results imply limited community structure in PA graphs at large $h$ and introduce methodological tools that may apply to broader random graph questions.

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 $μ$, 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, Pérez-Giménez, and Reiniger from 2019.

Theorems & Definitions (49)

• Definition 1: Modularity, NeGi04
• Theorem 2: prokhorenkova2017modularity_internet_Math, Theorem 4.2, Section 4.2
• Conjecture 3: prokhorenkova2017modularity_internet_Math
• Conjecture 4: prokhorenkova2017modularity_internet_Math
• Theorem 5
• Remark
• Corollary 6
• Remark
• Remark
• Definition 7: Measure $\mu$