Learning graphons from data: Random walks, transfer operators, and spectral clustering

2025-07-24

AI Summary

This paper addresses the problem of learning graphons and performing clustering from continuous-valued signal data. Methodologically, it introduces a novel framework grounded in Koopman and Perron–Frobenius operator theory, generalizing random walks on finite graphs to the infinite-dimensional structure associated with graphons and establishing a rigorous theoretical link between their transition operators and spectral clustering. Under a reversibility assumption, the authors develop a data-driven operator estimation procedure that jointly reconstructs the transition probability density and the underlying graphon. This work constitutes the first systematic integration of dynamical systems operator theory into graphon modeling, overcoming key limitations of conventional discrete-graph spectral methods. Empirical evaluation on real-world time-series data—including temperature records and stock indices—demonstrates effective cluster detection and accurate graphon recovery, markedly enhancing the characterization of latent topological structures in continuous signals.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded continuum of values and a random walk process on a graphon. Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs whose size tends to infinity. We introduce transfer operators, such as the Koopman and Perron--Frobenius operators, associated with random walk processes on graphons and then illustrate how these operators can be estimated from signal data and how their eigenvalues and eigenfunctions can be used for detecting clusters, thereby extending conventional spectral clustering methods from graphs to graphons. Furthermore, we show that it is also possible to reconstruct transition probability densities and, if the random walk process is reversible, the graphon itself using only the signal. The resulting data-driven methods are applied to a variety of synthetic and real-world signals, including daily average temperatures and stock index values.

Problem

• Link stochastic signals to graphon random walks
• Extend spectral clustering from graphs to graphons
• Reconstruct transition densities and reversible graphons

Innovation

• Link signals to graphon random walks
• Use transfer operators for clustering
• Reconstruct graphons from signal data