Influence of the seed in affine preferential attachment trees

Abstract

We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree $S$, vertices are attached one by one, each linked by an edge to a random vertex of the current tree, chosen with a probability proportional to an affine function of its degree. This yields a one-parameter family of preferential attachment trees $(T_{n}^{S})_{n\geq |S|}$, of which the linear model is a particular case. Depending on the choice of the parameter, the power-laws governing the degrees in $T_{n}^{S}$ have different exponents. We study the problem of the asymptotic influence of the seed $S$ on the law of $T_{n}^{S}$. We show that, for any two distinct seeds $S$ and $S'$, the laws of $T_{n}^{S}$ and $T_{n}^{S'}$ remain at uniformly positive total-variation distance as $n$ increases. This is a continuation of Curien et al. (J. Éc. Polytech. Math. 2 (2015) 1–34), which in turn was inspired by a conjecture of Bubeck et al. (IEEE Trans. Netw. Sci. Eng. 2 (2015) 30–39). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.