Clustering and Cliques in P.A random graphs with edge insertion

TL;DR

This paper studies the properties of clustering and clique numbers in a preferential attachment graph model with edge additions between existing vertices, establishing concentration inequalities and an inverse relation between these metrics.

Contribution

It introduces a model with dynamic edge addition, derives concentration bounds for clustering and clique numbers, and reveals an inverse relationship between these two graph statistics.

Findings

Concentration inequalities for clustering coefficient and clique number.

Inverse relation between clique number and global clustering coefficient.

Analysis under regularly varying function for edge addition probability.

Abstract

In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step $t$, either a new vertex is added with probability $f(t)$, or an edge is added between two existing vertices with probability $1-f(t)$. We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that $f(t)$ is a regularly varying function at infinity with index of regular variation $-\gamma$, where $\gamma \in [0,1)$. We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.