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

arXiv: https://arxiv.org/abs/2507.18147 · HTML: https://mlsanity.com/paper/2507.18147/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 67 references that are in the map.
- First posted on arXiv 24 Jul 2025.

## Builds on (most cited first, 21 of 21)

- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [A Consistent Histogram Estimator for Exchangeable Graph Models](https://mlsanity.com/paper/1402.1888/) (2014)
- [Graphon Signal Processing](https://mlsanity.com/paper/2003.05030/) (2020)
- [A graphon-signal analysis of graph neural networks](https://mlsanity.com/paper/2305.15987/) (2023)
- [Implicit Graphon Neural Representation](https://mlsanity.com/paper/2211.03329/) (2022)
- [Centrality measures for graphons: Accounting for uncertainty in networks](https://mlsanity.com/paper/1707.09350/) (2017)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Graph and graphon neural network stability](https://mlsanity.com/paper/2010.12529/) (2020)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Graphon Control of Large-scale Networks of Linear Systems](https://mlsanity.com/paper/1807.03412/) (2018)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Consensus Formation in First-Order Graphon Models with Time-Varying Topologies](https://mlsanity.com/paper/2111.03900/) (2021)
- [Transfer operators on graphs: Spectral clustering and beyond](https://mlsanity.com/paper/2305.11766/) (2023)
- [Transferability of Graph Neural Networks using Graphon and Sampling Theories](https://mlsanity.com/paper/2307.13206/) (2023)
- [Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization](https://mlsanity.com/paper/2303.01483/) (2023)
- [Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian](https://mlsanity.com/paper/2407.12864/) (2024)

## Built on by (most cited first, 2 of 2)

- [Optimization of randomized neural networks for transfer operator approximation](https://mlsanity.com/paper/2605.23689/) (2026)
- [Functional dynamic mode decomposition: Learning infinite-dimensional systems from data](https://mlsanity.com/paper/2609.29159/) (2026)

## Abstract (arXiv)

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.