Scale-Free Property for Degrees and Weights in an N-Interactions Random Graph Model*

Abstract

A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of N vertices (N ≥ 3). A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of the interactions of the vertex. The asymptotic behaviour of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2, ∞) can be achieved. The proofs are based on discrete time martingale theory.

Weights of Cliques in a Random Graph Model Based on Three-Interactions*

“…We can see that Theorem 2.2 is a particular case of Theorem 4.1 for N = 3. We remark that in the general N -interactions model the power law distributions both for the weights and degrees of vertices are known (see [8] and [9]).…”

“…Throughout this section we shall study the following N -interactions model (see [9]). For the sake of brevity, a complete graph with m vertices we call an m-clique.…”

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

“…Scale-free weight distributions, both for the edges and the triangles in the three-interactions model, were obtained in [15]. Instead of the three-interactions model, an evolution rule based on interactions of N vertices (N 3 fixed) was studied in [16].…”

“…Throughout this paper we shall study the following N -interactions model (defined in [16]). Here N 3 is a fixed integer.…”

“…Power law degree and weight distributions for vertices in the general N -interactions model were obtained in [16,17]. The asymptotic behaviour of the weights of the N -cliques was examined and power law weight distribution for the N -cliques was obtained in [15].…”