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 Xn(H, W) the number of copies of H in a W-random graph on n vertices. The asymptotic distribution of Xn(H, W) was recently obtained by Hladk\'y, Pelekis, and Sileikis (2021) 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)|-12. On the other hand, if W is H-regular, then the fluctuations are of order n|V(H)|-1 and the limiting distribution of Xn(H, W) can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centered 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 generalized U-statistics developed by Janson and Nowicki (1991). 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, Xn(H, W) has a degenerate limit even under the scaling n|V(H)|-1. We give an example of this degeneracy with H=K1, 3 (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher-order degeneracies.