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

TL;DR

This paper studies a random graph model where vertices, edges, and triangles gain weights through three-vertex interactions, showing that their weight distributions follow power laws asymptotically, with proofs using martingale techniques.

Contribution

It proves that the weight distributions of edges and triangles in the model are asymptotically power laws, extending known results for vertex weights.

Findings

Vertex weights follow a power law distribution.

Edge and triangle weights also follow power laws asymptotically.

Numerical results support the theoretical findings.

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.