Cut norm discontinuity of triangular truncation of graphons

TL;DR

This paper demonstrates that the triangular truncation operator on $L^p$ graphons is discontinuous in the cut norm, with the operator's norm growing unboundedly as matrix size increases, revealing limitations in graphon truncation methods.

Contribution

The paper proves the discontinuity of the triangular cut operator on $L^p$ graphons and shows the unbounded growth of the operator norm on symmetric matrices, extending to the graphon setting.

Findings

Triangular truncation operator norm grows to infinity with matrix size.

Discontinuity of the cut norm for the triangular truncation operator.

Unboundedness of the operator on symmetric matrices and graphons.

Abstract

The space of $L^p$ graphons, symmetric measurable functions $w: [0,1]^2 \to \mathbb{R}$ with finite $p$-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator $M_{\chi}$ acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as $n\to \infty$, the norm of the triangular truncation operator $\mathcal{T}_n$ on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of $L^p$ graphons, the norm growth of $\mathcal{T}_n$ generalizes to the unboundedness of $M_{\chi}$. We also show that the norm of $\mathcal{T}_n$ grows to infinity on symmetric matrices equipped with the operator norm.