Local properties of random mappings with exchangeable in-degrees

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=lⁿ 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_nD^ 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.