# A limit theorem for small cliques in inhomogeneous random graphs

## TL;DR

This paper establishes a limit theorem for the number of small cliques in inhomogeneous random graphs generated by graphons, revealing different fluctuation regimes and their distributions, including normal and chi-square types.

## Contribution

It introduces a new limit theorem for clique counts in W-random graphs, connecting fluctuations to spectral properties of the graphon, with alternative proofs linked to graphon theory.

## Key findings

- Fluctuations can be of order 0, n^{r-1}, or n^{r-0.5}.
- When fluctuations are of order n^{r-0.5}, they are normal.
- Order n^{r-1} fluctuations can be normal or chi-square.

## Abstract

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erd\H{o}s--R\'enyi random graph, called $W$-random graphs. We prove, via the method of moments, a limit theorem for the number of $r$-cliques in such random graphs. We show that, whereas in the case of dense Erd\H{o}s--R\'enyi 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 $W$. These results can also be deduced from a general setting [Janson and Nowicki, PTRF 1991], based on the projection method. In addition to providing alternative proofs, our approach makes direct links to the theory of graphons.