### Abstract

In this article we investigate the asymptotic structure of a random mapping model with preferential attachment, T_{n}^{?}, which maps the set {1,2,. . . , n} into itself. The model T_{n}^{?} was introduced in a companion article [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105-126] and the asymptotic structure of the associated directed graph G_{n}^{?} which represents the action of T_{n}^{?} on the set {1,2, . . . , n} was investigated in [Hansen and Jaworski, Random Struct Algorithms 33 (2008), 105-126] and [Hansen and Jaworski, Adv Appl Probab 40 (2008), 183-205] in the case when the attraction parameter ? > 0 is fixed as n ? 8. In this article we consider the asymptotic structure of G_{n}^{?} when the attraction parameter ? = ?(n) is a function of n as n ? 8. We show that there are three distinct regimes during the evolution of G _{n}^{?} (i) ?n ? 8 as n ? 8, (ii)?n ? ß > 0 as n ? 8, and (iii) ?n ? 0 as n ? 8. It turns out that the asymptotic structure of G _{n}^{?} 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. © 2008 Wiley Periodicals, Inc.

Original language | English |
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Pages (from-to) | 87-111 |

Number of pages | 25 |

Journal | Random Structures and Algorithms |

Volume | 34 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

Event | 13th International Conference on Random Structures and Algorithms - Tel Aviv, Israel Duration: 28 May 2007 → 1 Jun 2007 |

### Keywords

- Component structure
- Evolution
- Exchangeable in-degrees
- Preferential attachment
- Random mappings

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## Cite this

*Random Structures and Algorithms*,

*34*(1), 87-111. https://doi.org/10.1002/rsa.20251