Cut norm discontinuity of triangular truncation of graphons

Abstract

The space of Lp graphons, symmetric measurable functions w:[0,1]2→R with finite p-norm, features heavily in the study of sparse graph limit theory. We show that the triangular cut operator Mχ—the operator that sets all values of a graphon below the main diagonal of the unit square to 0—acting on this space is not continuous with respect to the cut norm. This is achieved by showing that as n→∞, the norm of the triangular truncation operator Tn on symmetric matrices equipped with the cut norm grows to infinity as well. Due to the density of symmetric matrices in the space of Lp graphons, the norm growth of Tn generalizes to the unboundedness of Mχ. We also show that the norm of Tn grows to infinity on symmetric matrices equipped with the operator norm.