The asymptotic behaviour of the weights and the degrees in an   N-interactions random graph model
  A random graph evolution based on the interactions of N vertices is studied. During the evolution both the preferential attachment method and the uniform choice of vertices are allowed. The weight of a vertex means the number of its interactions. The asymptotic behaviour of the weight and the degree of a fixed vertex, moreover the limit of the maximal weight and the maximal degree are described. The proofs are based on martingale methods. 
Istv\'an Fazekas, Bettina Porv\'azsnyik
math.PR