Asymptotics for cliques in scale-free random graphs

Abstract: In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung--Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities $h<sup>{1-\alpha}l(h)$, where $\alpha&gt;2$ and $l$ is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all non-integer $\alpha &gt; 2$. We also explain why the case of an integer $\alpha$ is different, and present partial results for the asymptotics in that case.