A central limit theorem for singular graphons

TL;DR

This paper studies the properties of singular graphons, which have unusually low variance in subgraph densities, and shows they share many features with Erdős-Rényi graphs, including a generalized central limit theorem.

Contribution

It investigates the properties of singular graphons, conjectures they are only constant graphons, and extends the CLT to these cases, linking spectral properties to the conjecture.

Findings

Singular graphons have variance of subgraph densities of order O(n^{-2})

Scaled densities in singular graphons converge in distribution, generalizing the CLT for Erdős-Rényi graphs

Spectral equations for Laplacian characteristic polynomials are established

Abstract

We associate to a graphon $\gamma$ the sequence of $W$-random graphs $(G_n(\gamma))_{n \geq 1}$. We say that the graphon is singular if, for any finite graph $F$, the homomorphism density $t(F,G_n(\gamma))$ has a variance of order $O(n^{-2})$. This behavior is singular because generically, the density of a fixed finite graph $F$ in a $W$-random graph has a variance of order $O(n^{-1})$. We conjecture that the only singular graphons are the constant graphons $\gamma_p$ with $p \in [0,1]$, corresponding to the Erd\H{o}s-R\'enyi random graphs $G(n,p)$. In this paper, we investigate the general properties of the singular graphons, and we show that they share many properties with the Erd\H{o}s-R\'enyi random graphs. In particular, if $\gamma$ is a singular graphon, then the scaled densities $n(t(F,G_n(\gamma))-\mathbb{E}[t(F,G_n(\gamma))])$ converge in joint distribution. This generalises the central limit theorem satisfied by the Erd\H{o}s-R\'enyi random graphs $G(n,p)$; however, the limiting distribution might be non-Gaussian if the conjecture does not hold. We also establish an equation satisfied by the characteristic polynomial of the Laplacian of the graph $G_n(\gamma)$ associated to a singular graphon; this opens the way to a spectral approach of the conjecture.