A random graph evolution based on interactions ofNvertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of anM-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixedM-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.