Predecessors and successors in random mappings with exchangeable in-degrees (Q2854077)

The authors consider a random mapping model in which they determine the distributions of the number of predecessors and the number of successors of a given set of vertices. In this random mapping model, the in-degree sequence is exchangeable. They derive exact formulae for these distributions and show that their expected values can be given in terms of the distributions of simple functions of the in-degree variables. As an application of these results, they consider two variations of the model which correspond to random mappings with preferential and anti-preferential attachment. There, they determine the exact distributions for the number of predecessors and the number of successors in these cases. They also characterise, for these two cases, the asymptotic behaviour of the expected numbers of predecessors and successors. Subsequently, these results are interpreted in terms of the critical behaviour of epidemic processes on random mapping graphs.