# Learning Time-Varying Graphs from Online Data

arXiv: https://arxiv.org/abs/2110.11017 · HTML: https://mlsanity.com/paper/2110.11017/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 21 Oct 2021.

## Builds on (most cited first, 16 of 16)

- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Network Inference via the Time-Varying Graphical Lasso](https://mlsanity.com/paper/1703.01958/) (2017)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Time-Varying Convex Optimization: Time-Structured Algorithms and Applications](https://mlsanity.com/paper/2006.08500/) (2020)
- [Online Topology Inference from Streaming Stationary Graph Signals with Partial Connectivity Information](https://mlsanity.com/paper/2007.03653/) (2020)
- [A Class of Prediction-Correction Methods for Time-Varying Convex Optimization](https://mlsanity.com/paper/1509.05196/) (2015)
- [Advances in Distributed Graph Filtering](https://mlsanity.com/paper/1808.03004/) (2018)
- [Online Discriminative Graph Learning from Multi-Class Smooth Signals](https://mlsanity.com/paper/2101.00184/) (2021)
- [Time-Varying Graph Learning with Constraints on Graph Temporal Variation](https://mlsanity.com/paper/2001.03346/) (2020)
- [Online Topology Identification from Vector Autoregressive Time Series](https://mlsanity.com/paper/1904.01864/) (2019)
- [Online Non-linear Topology Identification from Graph-connected Time Series](https://mlsanity.com/paper/2104.00030/) (2021)
- [Tracking Switched Dynamic Network Topologies from Information Cascades](https://mlsanity.com/paper/1606.08882/) (2016)
- [Online Time-Varying Topology Identification via Prediction-Correction Algorithms](https://mlsanity.com/paper/2010.11634/) (2020)

## Built on by (most cited first, 12 of 12)

- [Sparse Covariance Neural Networks](https://mlsanity.com/paper/2410.01669/) (2024)
- [Structural Knowledge Informed Continual Multivariate Time Series Forecasting](https://mlsanity.com/paper/2402.12722/) (2024)
- [Online Learning Of Expanding Graphs](https://mlsanity.com/paper/2409.08660/) (2024)
- [Time-Varying Graph Learning for Data with Heavy-Tailed Distribution](https://mlsanity.com/paper/2501.00606/) (2024)
- [Online Network Inference from Graph-Stationary Signals with Hidden Nodes](https://mlsanity.com/paper/2409.08760/) (2024)
- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)
- [When GNNs Fail: Quantifying and Overcoming Temporal Correlation Volatility in Time Series](https://mlsanity.com/paper/2608.07333/) (2026)
- [Online Proximal ADMM for Graph Learning from Streaming Smooth Signals](https://mlsanity.com/paper/2409.12916/) (2024)
- [Co-Hub Node Based Multiview Graph Learning with Theoretical Guarantees](https://mlsanity.com/paper/2512.12435/) (2025)
- [Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization](https://mlsanity.com/paper/2605.18316/) (2026)
- [Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation](https://mlsanity.com/paper/2606.24011/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)

## Abstract (arXiv)

This work proposes an algorithmic framework to learn time-varying graphs from online data. The generality offered by the framework renders it model-independent, i.e., it can be theoretically analyzed in its abstract formulation and then instantiated under a variety of model-dependent graph learning problems. This is possible by phrasing (time-varying) graph learning as a composite optimization problem, where different functions regulate different desiderata, e.g., data fidelity, sparsity or smoothness. Instrumental for the findings is recognizing that the dependence of the majority (if not all) data-driven graph learning algorithms on the data is exerted through the empirical covariance matrix, representing a sufficient statistic for the estimation problem. Its user-defined recursive update enables the framework to work in non-stationary environments, while iterative algorithms building on novel time-varying optimization tools explicitly take into account the temporal dynamics, speeding up convergence and implicitly including a temporal-regularization of the solution. We specialize the framework to three well-known graph learning models, namely, the Gaussian graphical model (GGM), the structural equation model (SEM), and the smoothness-based model (SBM), where we also introduce ad-hoc vectorization schemes for structured matrices (symmetric, hollows, etc.) which are crucial to perform correct gradient computations, other than enabling to work in low-dimensional vector spaces and hence easing storage requirements. After discussing the theoretical guarantees of the proposed framework, we corroborate it with extensive numerical tests in synthetic and real data.