A random mapping with preferential attachment

In this paper we investigate the asymptotic structure of a random mapping model with preferential attachment, T ρ n , which maps the set {1, 2, ..., n} into itself. The model T ρ n was introduced in a companion paper [11] and the asymptotic structure of the associated directed graph Gn which represents the action of T ρ n on the set {1, 2, ..., n} was investigated in [11] and [12] in the case when the attraction parameter ρ > 0 is fixed as n → ∞. In this paper we consider the asymptotic structure of Gn when the attraction parameter ρ ≡ ρ(n) is a function of n as n →∞. We show that there are three distinct regimes during the evolution of Gn: (i) ρn → ∞ as n → ∞, (ii) ρn → β > 0 as n →∞, and (iii) ρn → 0 as n →∞. It turns out that the asymptotic structure of Gn is, in some cases, quite different from the asymptotic structure of well-known models such as the uniform random mapping model and models with an attracting center. In particular, in regime (ii) we obtain some interesting new limiting distributions which are related to the incomplete gamma function.