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.

A population evolution model and its applications to random networks

“…In this section we apply the theorems of Section 2 to the weights of the M-cliques of the N-interactions random graph model (see [9], [10], [11]). First we recall that, by the usual definition, a complete graph with M vertices is called an M-clique.…”

“…The N-interactions model for N = 3 was introduced in Backhausz and Móri [2] (see also [3]). The general N-interactions model introduced and studied in Fazekas and Porvázsnyik [9], [10], [11]. The model incorporates the preferential attachment rule and the uniform choice of vertices.…”

Taylor’s power law for the <italic>N</italic>-stars network evolution model

“…In both cases the terminal vertices can be chosen either uniformly or according to the preferential attachment rule. In [1,13] and [12] the ideas of Cooper and Frieze [8] were applied, but instead of the original preferential attachment rule, the terminal vertices were chosen according to the weights of certain cliques.…”