Monotone graph limits and quasimonotone graphs

TL;DR

This paper establishes a connection between the monotonicity of graphons and a quasi-monotonicity property of graph sequences, providing new inequalities relating cut and $L^1$ norms for monotone kernels.

Contribution

It characterizes monotone graphons via a quasi-monotonicity property of graph sequences and proves a novel inequality relating cut and $L^1$ norms for monotone kernels.

Findings

Kernel is monotone iff the sequence satisfies a quasi-monotonicity property.

Proved a new inequality relating cut and $L^1$ norms for monotone kernels.

No such inequality holds for general kernels.

Abstract

The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences $(G_n)$ of graphs in terms of a limiting object which may be represented by a symmetric function $W$ on $[0,1]$, i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a `quasi-monotonicity' property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and $L^1$ norms of kernels of the form $W_1-W_2$ with $W_1$ and $W_2$ monotone that may be of interest in its own right; no such inequality holds for general kernels.