On derivatives of graphon parameters

TL;DR

This paper provides an elementary proof of a theorem characterizing graphon parameters with vanishing derivatives as linear combinations of homomorphism densities, clarifying their structure in graphon space.

Contribution

It offers a simplified proof of a key theorem relating derivatives of graphon parameters to homomorphism densities, enhancing understanding of their structure.

Findings

Any graphon parameter with vanishing (N+1)-th derivatives is a linear combination of homomorphism densities.

The proof simplifies previous arguments, making the theorem more accessible.

Clarifies the structure of differentiable graphon parameters in graph theory.

Abstract

We give a short elementary proof of the main theorem in the paper "Differential calculus on graphon space" by Diao et al. (JCTA 2015), which says that any graphon parameters whose $(N+1)$-th derivatives all vanish must be a linear combination of homomorphism densities $t(H, -)$ over graphs $H$ on at most $N$ edges.