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

TL;DR

This paper develops methods to infer and analyze the underlying stochastic processes of signals evolving over time by linking them to random walks on graphons, enabling clustering and reconstruction from data.

Contribution

It introduces transfer operators for graphons, estimates their spectral properties from data, and extends spectral clustering and graphon reconstruction techniques to real-world signals.

Findings

Eigenvalues and eigenfunctions can be estimated from data.

Spectral clustering on graphons can detect signal clusters.

Transition densities and graphons can be reconstructed from signals.

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.