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Scalable Implicit Graphon Learning

Ali Azizpour, Nicolas Zilberstein, Santiago Segarra

TL;DR

SIGL introduces a scalable, resolution-free graphon learning framework by marrying implicit neural representations with graph neural networks to align nodes across multiple graphs and replace GW-based losses with an efficient MSE objective on histogram observations. The method yields a continuous graphon representation f_θ: [0,1]^2 → [0,1] that can be sampled at arbitrary resolutions and extends to learning parametric families ω_α via an augmented input. Theoretical guarantees establish asymptotic consistency as expressivity grows, and extensive experiments show SIGL outperforms state-of-the-art baselines in both estimation accuracy and scalability, while enabling effective graph data augmentation through graphon mixup. The practical impact lies in scalable graphon estimation for large networks and improved performance in downstream tasks that rely on realistic graph generation and augmentation.

Abstract

Graphons are continuous models that represent the structure of graphs and allow the generation of graphs of varying sizes. We propose Scalable Implicit Graphon Learning (SIGL), a scalable method that combines implicit neural representations (INRs) and graph neural networks (GNNs) to estimate a graphon from observed graphs. Unlike existing methods, which face important limitations like fixed resolution and scalability issues, SIGL learns a continuous graphon at arbitrary resolutions. GNNs are used to determine the correct node ordering, improving graph alignment. Furthermore, we characterize the asymptotic consistency of our estimator, showing that more expressive INRs and GNNs lead to consistent estimators. We evaluate SIGL in synthetic and real-world graphs, showing that it outperforms existing methods and scales effectively to larger graphs, making it ideal for tasks like graph data augmentation.

Scalable Implicit Graphon Learning

TL;DR

SIGL introduces a scalable, resolution-free graphon learning framework by marrying implicit neural representations with graph neural networks to align nodes across multiple graphs and replace GW-based losses with an efficient MSE objective on histogram observations. The method yields a continuous graphon representation f_θ: [0,1]^2 → [0,1] that can be sampled at arbitrary resolutions and extends to learning parametric families ω_α via an augmented input. Theoretical guarantees establish asymptotic consistency as expressivity grows, and extensive experiments show SIGL outperforms state-of-the-art baselines in both estimation accuracy and scalability, while enabling effective graph data augmentation through graphon mixup. The practical impact lies in scalable graphon estimation for large networks and improved performance in downstream tasks that rely on realistic graph generation and augmentation.

Abstract

Graphons are continuous models that represent the structure of graphs and allow the generation of graphs of varying sizes. We propose Scalable Implicit Graphon Learning (SIGL), a scalable method that combines implicit neural representations (INRs) and graph neural networks (GNNs) to estimate a graphon from observed graphs. Unlike existing methods, which face important limitations like fixed resolution and scalability issues, SIGL learns a continuous graphon at arbitrary resolutions. GNNs are used to determine the correct node ordering, improving graph alignment. Furthermore, we characterize the asymptotic consistency of our estimator, showing that more expressive INRs and GNNs lead to consistent estimators. We evaluate SIGL in synthetic and real-world graphs, showing that it outperforms existing methods and scales effectively to larger graphs, making it ideal for tasks like graph data augmentation.

Paper Structure

This paper contains 44 sections, 7 theorems, 44 equations, 11 figures, 3 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. Graphon.
  4. INR.
  5. SCALABLE LEARNING OF IMPLICIT GRAPHON NEURAL REPRESENTATIONS
  6. Scalable Learning of a Graphon
  7. Step 2: Histogram approximation.
  8. Step 3: INR training.
  9. SIGL for learning a parametric family of graphons.
  10. Asymptotic consistency of SIGL
  11. EXPERIMENTS
  12. Single graphon estimation
  13. Setup.
  14. Baselines.
  15. Performance.
  16. ...and 29 more sections

Key Result

Proposition 1

Let $\omega(x,y)\colon [0, 1]^2 \rightarrow [0,1]$ be the true graphon and $f_\theta(x,y)\colon [0, 1]^2 \rightarrow [0,1]$ be the estimated graphon by SIGL. Assuming that $\omega$ is Lipschitz continuous with constant ${\mathcal{L}}>0$, and selecting the pooling window $h = \log n$ in Step 2, then, where $\varepsilon_{tr}$ is the maximum regression error of the INR on the training data points of

Figures (11)

  • Figure 1: Overview of SIGL. (a) A set of graphs is observed from the true (unknown) graphon. (b) Step 1: A GNN-INR module learns the latent variables of the nodes in the observed graphs, enabling alignment of the graphs. (c) Step 2: We build the dataset ${\mathcal{C}}$ containing the upper triangular part the histogram approximations in the dataset ${\mathcal{I}}$. (d) Step 3: An INR is trained using the coordinates and histogram values to estimate the graphon. The learned INR can be sampled at any arbitrary resolution $R$ to generate the final estimated graphon.
  • Figure 2: Single graphon estimation. (a) SIGL, compared to the other graphon estimation methods, demonstrates superior performance. (b) Increasing the number of nodes helps to have a better estimation of the graphon. Also, our method still outperforms IGNR. (c) SIGL estimates graphons faster than IGNR, especially in larger graphs.
  • Figure 3: Visualization of SIGL for: (a) a monotonic graphon, where the GNN learns latent variables that decrease as the node degree increases, and (b) a non-monotonic SBM graphon, where the GNN assigns similar latent variables to nodes within the same block, effectively seperating the block structure.
  • Figure 4: SIGL performs similarly to single graphon estimation when estimating the graphon in the parameterized scenario. (a) Parametrized monotnic graphon. (b) Parametrized SBM (non-monotonic) graphon.
  • Figure 5: Latent space for the graphs are distinct for different true values of $\alpha$.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Proposition 1: MSE of the estimator
  • Lemma 1: Piecewise Constant Function Approximation
  • Lemma 2: Bounds on $\lVert H - H^{\omega_r} \rVert_2^2$
  • Lemma 3: Bounds on $\lVert \hat{H} - H \rVert_2^2$
  • Lemma 4: Bounds on $\lVert \hat{\omega}^{est}_k - \hat{H} \rVert_2^2$
  • Lemma 5: Bounds on different resolutions of the estimated graphon
  • Lemma 6: MSE of sampled estimation