Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

Abstract

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.

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]).…”

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

“…The results of this paper will be based on it. The particular cases N = 3 and N = 4 are included in [7] and [13], respectively. Let…”

“…The power law degree distribution in that model was proved in [7]. In [13], instead of the three-interactions model, interactions of four vertices were studied. It turned out that in the seemingly complicated four-interactions model the asymptotic behaviour is as simple as in the three-interactions model.…”

“…In this paper, we extend the model and the results of [6], [7] and [13] to interactions of N vertices. Our model is the following.…”

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

“…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].…”