Graph limits: An alternative approach to s-graphons

TL;DR

This paper introduces an alternative approach to s-graphon convergence by characterizing it through the convergence of specific compact sets of probability measures, providing a new perspective on graph limits.

Contribution

It establishes the equivalence between s-convergence of graph sequences and the convergence of shape sets of measures, offering a novel characterization similar to graphon convergence.

Findings

s-convergence is equivalent to shape set convergence

Shape sets are compact and characterize graph limits

Provides a new framework for understanding graph limits

Abstract

We show that s-convergence of graph sequences is equivalent to the convergence of certain compact sets, called shapes, of Borel probability measures. This result is analogous to the characterization of graphon convergence (with respect to the cut distance) by the convergence of envelopes, due to Dole\v{z}al, Greb\'{i}k, Hladk\'{y}, Rocha, and Rozho\v{v}.