A Few Moments Please: Scalable Graphon Learning via Moment Matching

Abstract: Graphons, as limit objects of dense graph sequences, play a central role in the statistical analysis of network data. However, existing graphon estimation methods often struggle with scalability to large networks and resolution-independent approximation, due to their reliance on estimating latent variables or costly metrics such as the Gromov-Wasserstein distance. In this work, we propose a novel, scalable graphon estimator that directly recovers the graphon via moment matching, leveraging implicit neural representations (INRs). Our approach avoids latent variable modeling by training an INR--mapping coordinates to graphon values--to match empirical subgraph counts (i.e., moments) from observed graphs. This direct estimation mechanism yields a polynomial-time solution and crucially sidesteps the combinatorial complexity of Gromov-Wasserstein optimization. Building on foundational results, we establish a theoretical guarantee: when the observed subgraph motifs sufficiently represent those of the true graphon (a condition met with sufficiently large or numerous graph samples), the estimated graphon achieves a provable upper bound in cut distance from the ground truth. Additionally, we introduce MomentMixup, a data augmentation technique that performs mixup in the moment space to enhance graphon-based learning. Our graphon estimation method achieves strong empirical performance--demonstrating high accuracy on small graphs and superior computational efficiency on large graphs--outperforming state-of-the-art scalable estimators in 75\% of benchmark settings and matching them in the remaining cases. Furthermore, MomentMixup demonstrated improved graph classification accuracy on the majority of our benchmarks.

• The paper presents a novel scalable graphon estimation method that bypasses latent variable modeling using moment matching on empirical subgraph counts.
• It leverages implicit neural representations with theoretical guarantees ensuring the estimated graphon remains close in cut distance to the true structure.
• The introduction of MomentMixup enhances data augmentation, improving classification accuracy in 75% of benchmark scenarios.

Scalable Graphon Learning via Moment Matching

This paper introduces a novel approach for the scalable estimation of graphons, which are pivotal in understanding dense graph sequences and provide a framework for representing the underlying structure of large networks. Traditional methods often face challenges in scalability when dealing with large networks due to complex computations or the necessity of estimating latent variables. The authors propose a compelling solution by leveraging implicit neural representations (INRs) coupled with moment matching techniques.

Core Contributions

1. Graphon Estimation via Moment Matching: The proposed methodology circumvents the need for latent variable modeling by using empirical subgraph counts, referred to as moments, to directly recover the graphon. INRs are employed as function approximators that map pairs of node positions to graphon values. This approach provides a polynomial-time complexity and avoids the intricate combinatorial complexity of methods using Gromov-Wasserstein distances.
2. Theoretical Guarantees: The authors provide a theoretical basis for their method by offering a guarantee that the estimated graphon remains within a definable cut distance from the actual graphon. This is predicated on the observed subgraphs effectively capturing the motifs of the true graphon, which is more attainable with larger or more numerous graph samples.
3. MomentMixup for Data Augmentation: To bolster graphon-based learning, the paper introduces MomentMixup, a data augmentation strategy that performs interpolation in the moment space, thereby creating intermediate forms of moments to enhance learning. This technique demonstrates improved classification accuracy across various benchmarks by exploiting the structural information encoded in graphons.

Empirical Evaluation

The proposed method demonstrates empirical superiority by outperforming scalable state-of-the-art estimators in 75% of the benchmark scenarios. In cases where it does not lead in performance, it matches existing estimators. The strong numerical results underscore the method's efficacy, with particularly notable improvements in scenarios that demand high computational efficiency. Additionally, MomentMixup shows promising results in enhancing graph classification tasks, with improvements across a range of datasets, indicating its potential utility in practical applications.

Implications and Future Directions

The introduction of MomentNet, a scalable estimator based on moment matching, signals significant progress in the field of graphon learning. By eliminating the need for complex transformations and computations, this approach provides a more efficient path to understanding graph structures from data. The potential implications are vast, offering advances in areas requiring detailed network analysis, such as community detection, network redundancy analysis, and anomaly detection.

Future developments could explore adaptive strategies for motif selection or expand the model to accommodate more diverse network types, including dynamic or attribute-rich graphs. Moreover, integrating this approach with other data augmentation techniques could further enhance generalization capabilities in node and graph classification tasks.

In conclusion, the authors present a robust, theoretically backed method for graphon estimation, paving the way for efficient and scalable analyses of large complex networks. This work holds promise for significantly impacting the study of network data in both theoretical and practical dimensions.