A new random mapping model

Abstract

We introduce a new random mapping model, $T_n^{\hat D}$, which maps the set $\{1,2,...,n\}$ into itself.The random mapping $T_n^{\hat D}$ is constructed using a collection of exchangeable random variables $\hat{D}_1, ....,\hat{D}_n$ which satisfy $\sum_{i=1}^n\hat{D}_i=n$. In the random digraph,$G_n^{\hat D}$, which represents the mapping $T_n^{\hat D}$, the in-degree sequence for the vertices is given by the variables $\hat{D}_1, \hat{D}_2, ..., \hat{D}_n$, and, in some sense,$G_n^{\hat D}$ 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 $\hat{D}_1, \hat{D}_2, ..., \hat{D}_n$. We also consider two special examples of $T_n^{\hat D}$ 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.