# Transfer operators on graphs: Spectral clustering and beyond

arXiv: https://arxiv.org/abs/2305.11766 · HTML: https://mlsanity.com/paper/2305.11766/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 46 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 7 of 7)

- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Hermitian matrices for clustering directed graphs: insights and applications](https://mlsanity.com/paper/1908.02096/) (2019)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Koopman-based spectral clustering of directed and time-evolving graphs](https://mlsanity.com/paper/2204.02951/) (2022)

## Built on by (most cited first, 3 of 3)

- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Learning graphons from data: Random walks, transfer operators, and spectral clustering](https://mlsanity.com/paper/2507.18147/) (2025)
- [Clustering Time-Evolving Networks Using the Spatio-Temporal Graph Laplacian](https://mlsanity.com/paper/2407.12864/) (2024)

## Abstract (arXiv)

Graphs and networks play an important role in modeling and analyzing complex interconnected systems such as transportation networks, integrated circuits, power grids, citation graphs, and biological and artificial neural networks. Graph clustering algorithms can be used to detect groups of strongly connected vertices and to derive coarse-grained models. We define transfer operators such as the Koopman operator and the Perron-Frobenius operator on graphs, study their spectral properties, introduce Galerkin projections of these operators, and illustrate how reduced representations can be estimated from data. In particular, we show that spectral clustering of undirected graphs can be interpreted in terms of eigenfunctions of the Koopman operator and propose novel clustering algorithms for directed graphs based on generalized transfer operators. We demonstrate the efficacy of the resulting algorithms on several benchmark problems and provide different interpretations of clusters.