Higher-order fluctuations in dense random graph models

TL;DR

This paper establishes quantitative bounds for the normal approximation of subgraph counts in dense random graphs generated by graphons, advancing the understanding of higher-order fluctuations in dense graph limit theory.

Contribution

It introduces a framework using generalised U-statistics and Gaussian Hilbert spaces to describe higher-order fluctuations and provides the first comprehensive central limit theorem in dense graph limit theory.

Findings

Quantitative bounds for multivariate normal approximation of subgraph counts

Identification of limiting Gaussian measures for fluctuations

Application of theory to network modeling

Abstract

Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they are key to understanding fluctuations of regular subgraph counts -- a cornerstone of dense graph limit theory. We also identify the resulting limiting Gaussian stochastic measures by means of the theory of generalised $U$-statistics and Gaussian Hilbert spaces, which we think is a suitable framework to describe and understand higher-order fluctuations in dense random graph models. With this article, we believe we answer the question "What is the central limit theorem of dense graph limit theory?". We complement the theory with some statistical applications to illustrate the use of centred subgraph counts in network modelling.