A limit theorem for small cliques in inhomogeneous random graphs

Abstract

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erdős-Rényi random graph, called W -random graphs. We obtain a limit theorem for the number of r-cliques in such random graphs. We show that, whereas in the case of dense Erdős-Rényi random graphs the fluctuations are normal of order n r−1 , the fluctuations in the setting of W -random graphs may be of order 0, n r−1 , or n r−0.5 . Furthermore, when the fluctuations are of order n r−0.5 they are normal, while when the fluctuations are of order n r−1 they exhibit either normal or a particular type of chi-square behavior whose parameters relate to spectral properties of W .

Central limit theorems for local network statistics

“…However, [22,Theorem 15.6] applies on the U -statistic component of B n , and we conjecture that nB n tends in distribution to a mixture of χ 2 distributions. Hladký et al [27,Theorem 1.1.c] shows that this conjecture holds at least for cliques. Proving this conjecture would require tighter bounds in (B.5) specific to this regime.…”