Transfer operators on random graph ensembles: Coherence, Clustering, and Phase transitions

We initiate a study of transfer operators—including Koopman, Perron–Frobenius, and their forward–backward compositions—defined over ensembles of random graphs, such as Erdős–Rényi graphs, configuration models, and directed stochastic block models. While classical spectral clustering relies on Laplacians of fixed graphs, and recent advances reinterpret clustering via transfer operators on deterministic networks, we propose a probabilistic operator-theoretic framework for analyzing clustering, coherence, and metastability in stochastic graph settings. Our first contribution is the formal definition and analysis of random transfer operators, viewed as operator-valued random matrices induced by random walks on graphs sampled from a distribution Gn. Extending deterministic constructions (e.g., Klus and Trower, (2024)), we study the spectral behavior of these operators and identify regimes in which their leading eigenfunctions localize on coherent sets—regions of the graph with internally consistent dynamics—emerging with high probability. In particular, we show that the forward–backward operator, a central tool in metastability analysis, exhibits spectral phase transitions as the underlying ensemble crosses structural thresholds (e.g., the percolation point p∼logn/n in Erdős–Rényi graphs). Our second contribution is algorithmic: we define Galerkin projections of these random operators based on finite trajectory data, and prove convergence in operator norm under mild mixing and sparsity assumptions. This leads to data-driven methods for detecting coherent structures, overlapping communities, and metastable regions in large, noisy, or partially observed networks. Finally, we formulate transfer operator spectral clustering on random graphs as a canonical correlation optimization problem, offering a principled connection to statistical learning and enabling coherent set recovery in directed, heterogeneous, or time-varying ensembles.