Fluctuations of subgraph counts in graphon based random graphs

Abstract

Given a graphon $W$ and a finite simple graph $H$ , with vertex set $V(H)$ , denote by $X_n(H, W)$ the number of copies of $H$ in a $W$ -random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$ . Towards this we introduce a notion of $H$ -regularity of graphons and show that if the graphon $W$ is not $H$ -regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$ . On the other hand, if $W$ is $H$ -regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$ . Our proofs use the asymptotic theory of generalised $U$ -statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$ -regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$ -regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$ . We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.