Fluctuations of subgraph counts in graphon based random graphs

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 Hladky, Pelekis, and Sileikis [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 Xn(H, W) has Gaussian fluctuations with scaling n(V(H)-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,K-3 (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.