Asymptotic diameter of preferential attachment model

TL;DR

The paper resolves the asymptotic diameter of the preferential attachment graph $G_n \sim \operatorname{PA}_n^{(m,\delta)}$ for $m\ge 2$ and $\delta>0$, showing $\operatorname{diam}(G_n)=(1+o(1))\log_\nu n$ with high probability, where $\nu$ is the exponential growth rate of the local weak limit. Building on the recent typical-distance result of VZ25, the authors develop a general framework that relates diameter to typical distance via a median-distance threshold $M_n$ and a small additive term, using a sprinkling argument and uniform growth of neighborhoods. They prove a key uniform-growth lemma ensuring $R_n=O((\log n)^{2/3})$ with $|N_{R_n}(v)|$ growing to exceed $(\log n)^4$ for all vertices, which bridges typical distance to diameter. The combination of these components yields the sharp diameter estimate and closes the open case in the parameter range, with potential applicability to a broader class of random graph models.

Abstract

We study the asymptotic diameter of the preferential attachment model $\operatorname{PA}\!_n^{(m,δ)}$ with parameters $m \ge 2$ and $δ> 0$. Building on the recent work \cite{VZ25}, we prove that the diameter of $G_n \sim \operatorname{PA}\!_n^{(m,δ)}$ is $(1+o(1))\log_νn$ with high probability, where $ν$ is the exponential growth rate of the local weak limit of $G_n$. Our result confirms the conjecture in \cite{VZ25} and closes the remaining gap in understanding the asymptotic diameter of preferential attachment graphs with general parameters $m \ge 1$ and $δ>-m$. Our proof follows a general recipe that relates the diameter of a random graph to its typical distance, which we expect to have applicability in a broader range of models.

Theorems & Definitions (15)

• Theorem 1.1
• Proposition 1.2: Theorem 1.1, VZ25
• Definition 1.3: Preferential attachment model
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 3.1
• Lemma 3.2
• proof