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

Abstract

A random graph evolution rule is considered. The graph evolution is based on interactions of three vertices. The weight of a clique is the number of its interactions. The asymptotic behaviour of the weights is described. It is known that the weight distribution of the vertices is asymptotically a power law. Here it is proved that the weight distributions both of the edges and the triangles are also asymptotically power laws. The proofs are based on discrete time martingale methods. Some numerical results are also presented.

Evolution of a Generalized Population Model

“…all weights of subcliques have a power-law distribution. Finally if we consider a special case when N = 3 and M = 2, we can obtain the same result as Theorem 2.3 in[2].…”

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

“…This result was presented in [15] for the case of M D 2, N D 3 and also for the case of M D N for arbitrary N > 2. The general case can be obtained by using the ideas of [18].…”

“…Therefore, as in [15], it is easy to show that the probability that the j th M -clique takes part in interaction at step…”

“…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]. Instead of the three-interactions model, an evolution rule based on interactions of N vertices (N 3 fixed) was studied in [16].…”

“…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].…”