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

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.