Limit theorems for the weights and the degrees in anN-interactions random graph model

Abstract

A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.

Keywords

• random graph
• preferential attachment
• scale-free
• power law
• submartingale
• 05c80
• 60g42