Uniqueness of Banach space valued graphons

TL;DR

This paper investigates the uniqueness of Banach space valued graphons, which are functions representing limits of multigraph sequences, establishing conditions under which these limits are uniquely determined by their generalized moments.

Contribution

It introduces a Carleman-type condition ensuring uniqueness of Banach space valued graphons and demonstrates cases where uniqueness fails, extending the theory of multigraph limits.

Findings

Uniqueness holds under a Carleman-type condition.

Non-uniqueness can occur even in real-valued cases.

Limits of multigraph sequences are uniquely determined when edge-distribution tails are small.

Abstract

A Banach space valued graphon is a function $W:(\Omega, \mathcal{A},\pi)^2\to\mathcal{Z}$ from a probability space to a Banach space with a separable predual, measurable in a suitable sense, and lying in appropriate $L^p$-spaces. As such we may consider $W(x,y)$ as a two-variable random element of the Banach space. A two-dimensional analogue of moments can be defined with the help of graphs and weak-* evaluations, and a natural question that then arises is whether these generalized moments determine the function $W$ uniquely -- up to measure preserving transformations. The main motivation comes from the theory of multigraph limits, where these graphons arise as the natural limit objects for convergence in a generalized homomorphism sense. Our main result is that this holds true under some Carleman-type condition, but fails in general even with $\mathcal{Z}=\mathbb{R}$, for reasons related to the classical moment-problem. In particular, limits of multigraph sequences are uniquely determined - up to measure preserving transformations - whenever the tails of the edge-distributions stay small enough.