Remarks on Graphons

TL;DR

This paper explores the structure of the set of graphons within a fuzzy subset semigroup, extending the classical graph limit theory by analyzing algebraic properties of graphons.

Contribution

It investigates the algebraic structure of graphons in a fuzzy subset semigroup, providing new insights into their composition and properties.

Findings

Characterization of graphons within the fuzzy subset semigroup

Analysis of the semigroup operation on graphons

Insights into the structure of graphon limits

Abstract

L. Lov\'asz and B. Szegedy proved in 2006 that the limits of convergent graph sequences can be described by measurable symmetric functions $W: [0, 1]\times [0, 1]\to [0, 1]$ called graphons. In our present paper we investigate the structure of the set of all graphons within the semigroup $(\mathfrak{F}([0, 1]^2); \circ)$ of all fuzzy subsets of the unit square $[0,1]^2=[0, 1]\times [0, 1]$, where the operation $\circ$ is defined by: for every $f, g\in \mathfrak{F}([0,1]^2)$ and every $s\in [0,1]^2$, $(f\circ g)(s)=\vee_{x\in [0,1]^2}(f(x)\wedge g(s))$.