Clustering and Cliques in Preferential Attachment 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). Finally, we also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient.