# Fluctuations of Subgraph Counts in Random Graphons

arXiv (Cornell University)（2021）

Abstract

Given a graphon $W$ and a finite simple graph $H$, 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 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 $X_n(H, W)$ has Gaussian fluctuations. On the other hand, if $W$ is $H$-regular, the limiting distribution of $X_n(H, W)$ can have a non-Gaussian component which is a (possibly) infinite weighted sum of centered chi-squared random variables, where the weights are 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).

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Key words

subgraph counts

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