## Abstract

In this article we introduce a new random mapping model, T_{n}^{D^}, which maps the set {1,2,..., n] into itself. The random mapping T_{n}^{D^} is constructed using a collection of exchangeable random variables D_{1},....,D_{n} which satisfy ?_{i=1}^{n} D^i,- = n. In the random digraph, G _{n}^{D^}, which represents the mapping T_{n}^{D^}, the in-degree sequence for the vertices is given by the variables D^_{1}, D^_{2},...,D^_{n}, and, in some sense, G_{n}^{D^} can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of D^_{1}, D^_{2} ,...,D^_{n}. We also consider two special examples of T_{n}^{D^} which correspond to random mappings with preferential and anti-preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. © 2007 Wiley Periodicals, Inc.

Original language | English |
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Pages (from-to) | 105-126 |

Number of pages | 22 |

Journal | Random Structures and Algorithms |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 2008 |

## Keywords

- Anti-preferential attachment
- Component structure
- Exchangeable in-degree sequence
- Preferential attachment
- Random mappings