## Abstract

In this paper we investigate the 'local' properties of a random mapping model, T_{n}^{D^}, which maps the set {1, 2,..., n} into itself. The random mapping T_{n}^{D^}, which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a colle tion of exchangeable random variables D^ _{l},..., D^_{n} which satisfy ? _{i=l}^{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. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G_{n}^{D^} - for example, the numbers of predecessors and successors of v in G_{n}^{D^}. We show that the distribution of several variables associated with the local structure of G_{n}^{D^} can be expressed in terms of expectations of simple 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, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest. © Applied Probability Trust 2008.

Original language | English |
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Pages (from-to) | 183-205 |

Number of pages | 23 |

Journal | Advances in Applied Probability |

Volume | 40 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2008 |

## Keywords

- Exchangeable in-degrees
- Local structure
- Random mapping