Moments of Two-Variable Functions and the Uniqueness of Graph Limits

TL;DR

This paper proves that the moments of symmetric functions on [0,1]^2 uniquely determine the function up to measure-preserving transformations, ensuring the uniqueness of graph limits for convergent dense graph sequences.

Contribution

It establishes that moments of symmetric functions uniquely identify the functions up to measure-preserving transformations, confirming the uniqueness of graph limits.

Findings

Functions are determined by their moments up to measure-preserving transformations

Limit of convergent dense graph sequences is unique up to measure-preserving transformation

Provides a moment-based characterization of graph limits

Abstract

For a symmetric bounded measurable function W on [0,1]^2, "moments" of W can be defined as values t(F,W) indexed by simple graphs. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. This implies that the limit of a convergent dense graph sequence is unique up to measure preserving transformation.