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A Short Note on Upper Bounds for Graph Neural Operator Convergence Rate

Roxanne Holden, Luana Ruiz

TL;DR

This note analyzes how graph neural operator spectra converge when finite graphs sampled from a graphon limit, enabling transferability guarantees. By combining graphon convergence in the cut distance with Weyl's inequality, it derives explicit eigenvalue-rate bounds under no assumptions, global Lipschitz, and piecewise-Lipschitz conditions: $|\lambda_i(W_{G_n})-\lambda_i(W)| = O(1/\sqrt[4]{\log n})$ in the standard case, $O(\sqrt{\log n / n})$ under Lipschitz, and $O((\log n / n)^{1/4})$ for piecewise-Lipschitz graphons. The bounds are shown to differ in permutation invariance and to yield distinct tradeoffs between generality and rate, with numerical experiments illustrating the bounds' behavior on synthetic and real data. The results provide a unified framework to assess GNN transferability and scalability for large graphs via graphon-based operator convergence.

Abstract

Graphons, as limits of graph sequences, provide a framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons yields operator-level convergence rates, enabling transferability analyses of GNNs. This note summarizes known bounds under no assumptions, global Lipschitz continuity, and piecewise-Lipschitz continuity, highlighting tradeoffs between assumptions and rates, and illustrating their empirical tightness on synthetic and real data.

A Short Note on Upper Bounds for Graph Neural Operator Convergence Rate

TL;DR

This note analyzes how graph neural operator spectra converge when finite graphs sampled from a graphon limit, enabling transferability guarantees. By combining graphon convergence in the cut distance with Weyl's inequality, it derives explicit eigenvalue-rate bounds under no assumptions, global Lipschitz, and piecewise-Lipschitz conditions: in the standard case, under Lipschitz, and for piecewise-Lipschitz graphons. The bounds are shown to differ in permutation invariance and to yield distinct tradeoffs between generality and rate, with numerical experiments illustrating the bounds' behavior on synthetic and real data. The results provide a unified framework to assess GNN transferability and scalability for large graphs via graphon-based operator convergence.

Abstract

Graphons, as limits of graph sequences, provide a framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons yields operator-level convergence rates, enabling transferability analyses of GNNs. This note summarizes known bounds under no assumptions, global Lipschitz continuity, and piecewise-Lipschitz continuity, highlighting tradeoffs between assumptions and rates, and illustrating their empirical tightness on synthetic and real data.
Paper Structure (12 sections, 7 theorems, 13 equations, 2 figures)

This paper contains 12 sections, 7 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary Definitions
  3. Graphons
  4. Cut norm and cut distance
  5. The graphon operator and Weyl's inequality
  6. Graph Neural Operator Convergence Rates
  7. Standard case
  8. Lipschitz case
  9. Piecewise-Lipschitz case
  10. Comparison and discussion
  11. Numerical examples
  12. Conclusion

Key Result

Lemma 1

Let ${\mathbf K}: [0,1]^2 \to [-1,1]$. Then,

Figures (2)

  • Figure 1: Piecewise-Lipschitz graphon with 200 nodes, $K=4$, and per-piece Lipschitz constant ${\mathcal{L}}_k \leq 4$.
  • Figure 2: Examples of graphons. Left: Synthetic graphon $\textbf{f} = x \cdot y$ with Lipschitz constant $0.0265$ and largest piecewise-Lipschitz constant $0.353$. Center: Cora graphon with Lipschitz constant $60.653$ and largest piecewise-Lipschitz constant $99.5$. Right: PubMed graphon with Lipschitz constant $58.534$ and largest piecewise-Lipschitz constant $199.24$.

Theorems & Definitions (10)

  • Lemma 1: Adapted from ruiz2021graphon
  • Lemma 2: Second Sampling Lemma
  • Proposition 3: Standard case
  • proof
  • Lemma 4: Adapted from transferabilityruiz
  • Proposition 5: Lipschitz case
  • proof
  • Lemma 6: Adapted from ruiz2021graphonAvella_Medina_2020
  • Proposition 7: Piecewise-Lipschitz case
  • proof