2026/07/01 by Yiming Chen
#math.PR
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 (du+α)(dv+α), where du and dv are the current degrees of u and v, and α>0. Let L1 denote the size of the largest component, and set mc:=(αn)/(2(α+1)). We prove that if m=mc(1-ε), ε=ε(n)→0, ε3 n→∞, then L1=(1+op(1))(2(α+2))/(α+1)ε-2log(ε3 n) for every fixed α>0. Moreover, the same asymptotic holds whenever α=α(n)→ a∈(0,∞]. In particular, the constant 2(α+2)/(α+1) converges to the Erdős--Rényi value 2 as α→∞. If m=\lfloor \frac n2(1-ε)\rfloor and αε→∞, then L1=(2+op(1))ε-2log(ε3 n). The subcritical asymptotics for \(L1\) 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.