Cut norm discontinuity of triangular truncation of 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.