# Stationary Graph Processes and Spectral Estimation

arXiv: https://arxiv.org/abs/1603.04667 · HTML: https://mlsanity.com/paper/1603.04667/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2016.

## Built on by (most cited first, 30 of 36)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Synthesizing Decentralized Controllers with Graph Neural Networks and Imitation Learning](https://mlsanity.com/paper/2012.14906/) (2020)
- [Learning Time-Varying Graphs from Online Data](https://mlsanity.com/paper/2110.11017/) (2021)
- [Joint Network Topology Inference via a Shared Graphon Model](https://mlsanity.com/paper/2209.08223/) (2022)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Fairness-Aware Graph Filter Design](https://mlsanity.com/paper/2303.11459/) (2023)
- [Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals](https://mlsanity.com/paper/2404.02621/) (2024)
- [Node-Variant Graph Filters in Graph Neural Networks](https://mlsanity.com/paper/2106.00089/) (2021)
- [Joint inference of multiple graphs with hidden variables from stationary graph signals](https://mlsanity.com/paper/2110.03666/) (2021)
- [Learning Product Graphs from Spectral Templates](https://mlsanity.com/paper/2211.02893/) (2022)
- [Fairness-aware Optimal Graph Filter Design](https://mlsanity.com/paper/2310.14432/) (2023)
- [Graphon-aided Joint Estimation of Multiple Graphs](https://mlsanity.com/paper/2202.05686/) (2022)
- [Learning the Structure of Large Networked Systems Obeying Conservation Laws](https://mlsanity.com/paper/2206.07083/) (2022)
- [Learning Graph ARMA Processes from Time-Vertex Spectra](https://mlsanity.com/paper/2302.06887/) (2023)
- [Joint Graph and Vertex Importance Learning](https://mlsanity.com/paper/2303.08552/) (2023)
- [LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees](https://mlsanity.com/paper/2305.01379/) (2023)
- [Online Network Inference from Graph-Stationary Signals with Hidden Nodes](https://mlsanity.com/paper/2409.08760/) (2024)
- [Online Graph Topology Learning via Time-Vertex Adaptive Filters: From Theory to Cardiac Fibrillation](https://mlsanity.com/paper/2411.01567/) (2024)
- [Adaptive posterior distributions for uncertainty analysis of covariance matrices in Bayesian inversion problems for multioutput signals](https://mlsanity.com/paper/2501.01148/) (2025)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Statistical Testing on Directed Graphs by Surrogate Data Generation](https://mlsanity.com/paper/2606.00758/) (2026)
- [Graph Filter Transfer via Probability Density Ratio Weighting](https://mlsanity.com/paper/2210.14633/) (2022)
- [Estimation of a Causal Directed Acyclic Graph Process using Non-Gaussianity](https://mlsanity.com/paper/2211.13800/) (2022)
- [Locally Stationary Graph Processes](https://mlsanity.com/paper/2309.01657/) (2023)
- [Sampling Transferable Graph Neural Networks with Limited Graph Information](https://mlsanity.com/paper/2410.16593/) (2024)
- [CoVariance Filters and Neural Networks over Hilbert Spaces](https://mlsanity.com/paper/2509.13178/) (2025)
- [A Generative Model for Controllable Feature Heterophily in Graphs](https://mlsanity.com/paper/2509.23230/) (2025)
- [Estimating Fair Graphs from Graph-Stationary Data](https://mlsanity.com/paper/2510.07536/) (2025)
- [Decorrelating the Future: Joint Frequency Domain Learning for Spatio-temporal Forecasting](https://mlsanity.com/paper/2603.04418/) (2026)
- [Directed Graph Topology Inference via Graph Filter Identification](https://mlsanity.com/paper/2606.27455/) (2026)

## Abstract (arXiv)

Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. The contribution in this paper is twofold. Firstly, we propose a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical definition in the time domain. Our definition requires that stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. We will show that this is equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform. Secondly, we analyze the properties of the power spectral density and propose a number of methods to estimate it. We start with nonparametric approaches, including periodograms, window-based average periodograms, and filter banks. We then shift the focus to parametric approaches, discussing the estimation of moving-average (MA), autoregressive (AR) and ARMA processes. Finally, we illustrate the power spectral density estimation in synthetic and real-world graphs.