Sharp Asymptotics for the Largest Component in the Subcritical Regime of Preferential Attachment Without Vertex Growth

Abstract

We study the size of the largest component in Pittel's preferential attachment process without vertex growth. Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+\alpha)(d_v+\alpha)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $\alpha>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\frac{\alpha n}{2(\alpha+1)}.$ We prove that if $m=m_c(1-\varepsilon), \varepsilon=\varepsilon(n)\to0, \varepsilon^3 n\to\infty,$ then \[ L_1=(1+o_p(1))\frac{2(\alpha+2)}{\alpha+1}\varepsilon^{-2}\log(\varepsilon^3 n) \] for every fixed $\alpha>0$. Moreover, the same asymptotic holds whenever $\alpha=\alpha(n)\to a\in(0,\infty]$. In particular, the constant $2(\alpha+2)/(\alpha+1)$ converges to the Erd\H{o}s--R\'enyi value $2$ as $\alpha\to\infty$. If $m=\left\lfloor \frac n2(1-\varepsilon)\right\rfloor$ and $\alpha\varepsilon\to\infty$, then \[ L_1=(2+o_p(1))\varepsilon^{-2}\log(\varepsilon^3 n). \] The subcritical asymptotics for \(L_1\) resolve the problem left open by Janson and Warnke. The upper bound argument relies on the fact that, after conditioning on the degree sequence, the graph can be treated through the corresponding configuration model, the lower bound follows from tree component asymptotics and a second moment argument.