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Consistent sampling of Paley-Wiener functions on graphons

Hartmut Führ, Mahya Ghandehari

TL;DR

This work develops a graphon-based counterpart to Shannon sampling for Paley-Wiener functions by extending averaging sampling from graphs to graphons and formulating consistency results with graphon convergence. It introduces partition-based local operators and sampling functionals on graphons, derives global inequalities that bound signal energy in terms of local samples, and defines Paley-Wiener spaces via graphon Laplacians. A central contribution is a convergence theorem ensuring sampling estimates remain uniform across graph sequences converging in cut-norm to a limit graphon, linking graph limit theory with robust signal processing on graphs. The results enable convergence-consistent, robust sampling schemes for graphon signals and pave the way for practical, limit-aware design in large networks, with open questions about weaker convergence and partition construction.

Abstract

We study sampling methods for Paley-Wiener functions on graphons, thereby adapting and generalizing methods initially developed for graphs to the graphon setting. We then derive conditions under which such a sampling estimate is consistent with graphon convergence.

Consistent sampling of Paley-Wiener functions on graphons

TL;DR

This work develops a graphon-based counterpart to Shannon sampling for Paley-Wiener functions by extending averaging sampling from graphs to graphons and formulating consistency results with graphon convergence. It introduces partition-based local operators and sampling functionals on graphons, derives global inequalities that bound signal energy in terms of local samples, and defines Paley-Wiener spaces via graphon Laplacians. A central contribution is a convergence theorem ensuring sampling estimates remain uniform across graph sequences converging in cut-norm to a limit graphon, linking graph limit theory with robust signal processing on graphs. The results enable convergence-consistent, robust sampling schemes for graphon signals and pave the way for practical, limit-aware design in large networks, with open questions about weaker convergence and partition construction.

Abstract

We study sampling methods for Paley-Wiener functions on graphons, thereby adapting and generalizing methods initially developed for graphs to the graphon setting. We then derive conditions under which such a sampling estimate is consistent with graphon convergence.

Paper Structure

This paper contains 7 sections, 6 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Notations and background
  3. Convergence of graphs, graphons and $w$-random graphs
  4. Cut norm and converging graph sequences
  5. Graphon operators
  6. Graphon signal sampling
  7. Sampling from converging graph sequences

Key Result

Theorem 1

Let $w\in {\cal W}_0$ be a graphon, and consider a partition $\{S_1,\ldots,S_k\}$ of $[0,1]$ into measurable subsets. For $j\in[k]$, let $w_j$ denote the restriction of $w$ to ${S_j\times S_j}$, and $L_j$ be the associated Laplacian operator on $L^2(S_j)$ defined similar to Equation eq:Laplacian. Fo Then, for every $f\in L^2[0,1]$ and for $\epsilon>0$, we have

Theorems & Definitions (12)

  • Theorem 1
  • Claim 2
  • Remark 3
  • Corollary 5
  • Definition 6
  • Corollary 7
  • Definition 8
  • Corollary 9
  • Definition 10
  • Lemma 11
  • ...and 2 more