Asymptotic diameter of preferential attachment model

Abstract

We study the asymptotic diameter of the preferential attachment model $\mathrm{PA}{}_n^{(m,\hat{I}´)}$ with parameters $m\ge 2$ and $\hat{I}´>0$. Building on the recent work \cite{VZ25}, we prove that the diameter of $G_n\sim \mathrm{PA}{}_n^{(m,\hat{I}´)}$ is $(1+o(1)){\log }_{\hat{I}}½n$ with high probability, where $\hat{I}½$ 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 $\hat{I}´>-m$. 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.

11 pages