Asymptotics for Cliques in Scale-Free Random Graphs

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 h1-αl(h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{1-\alpha }l(h)$$\end{document}, where α>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >2$$\end{document} 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 α>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 2$$\end{document}. We also explain why the case of an integer α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is different, and present partial results for the asymptotics in that case.