Weights and Degrees in a Random Graph Model Based on 3-Interactions

Abstract

Abstract. In a random graph model introduced in [1] we give the joint asymptotic distribution of weights and degrees and prove scale-free property for the model. Moreover, we determine the asymptotics of the maximal weight and the maximal degree.

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

“…Power law degree distribution in the three-interactions model was proved in [13,14]. The asymptotic behaviour of the weight and the degree of a fixed vertex, as well as the limits of the maximal weight and the maximal degree, were also described in [13,14]. Scale-free weight distributions, both for the edges and the triangles in the three-interactions model, were obtained in [15].…”

“…In that procedure both the preferential attachment rule and the uniform choice of vertices are allowed, moreover, new links can be created between old vertices. The evolution method introduced in [13] in some sense resembles the one in [12]. However, the main feature of the model applied in [13] is the interaction of three vertices.…”

“…The evolution method introduced in [13] in some sense resembles the one in [12]. However, the main feature of the model applied in [13] is the interaction of three vertices. Power law degree distribution in the three-interactions model was proved in [13,14].…”

“…However, the main feature of the model applied in [13] is the interaction of three vertices. Power law degree distribution in the three-interactions model was proved in [13,14]. The asymptotic behaviour of the weight and the degree of a fixed vertex, as well as the limits of the maximal weight and the maximal degree, were also described in [13,14].…”

“…Moreover, we shall also consider the limiting properties of the maximal weight and the maximal degree of vertices. In our proofs we follow some ideas of [13,14]. However, the combinatorial problems for general N are much more difficult than for N D 3.…”