# Directed Graph Topology Inference via Graph Filter Identification

arXiv: https://arxiv.org/abs/2606.27455 · HTML: https://mlsanity.com/paper/2606.27455/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 42 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 24 of 24)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [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)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Network Topology Inference from Spectral Templates](https://mlsanity.com/paper/1608.03008/) (2016)
- [Learning heat diffusion graphs](https://mlsanity.com/paper/1611.01456/) (2016)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Signal Processing on Graphs: Causal Modeling of Unstructured Data](https://mlsanity.com/paper/1503.00173/) (2015)
- [Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables](https://mlsanity.com/paper/2111.05588/) (2021)
- [Online Topology Inference from Streaming Stationary Graph Signals with Partial Connectivity Information](https://mlsanity.com/paper/2007.03653/) (2020)
- [Joint Inference of Multiple Graphs from Matrix Polynomials](https://mlsanity.com/paper/2010.08120/) (2020)
- [Accelerated Graph Learning from Smooth Signals](https://mlsanity.com/paper/2110.09677/) (2021)
- [Block Coordinate Descent on Smooth Manifolds: Convergence Theory and Twenty-One Examples](https://mlsanity.com/paper/2305.14744/) (2023)
- [Robust Graph Filter Identification and Graph Denoising from Signal Observations](https://mlsanity.com/paper/2210.08488/) (2022)
- [Online Discriminative Graph Learning from Multi-Class Smooth Signals](https://mlsanity.com/paper/2101.00184/) (2021)
- [CoLiDE: Concomitant Linear DAG Estimation](https://mlsanity.com/paper/2310.02895/) (2023)
- [Identifying the Topology of Undirected Networks from Diffused Non-stationary Graph Signals](https://mlsanity.com/paper/1801.03862/) (2018)
- [Tensor Decompositions for Identifying Directed Graph Topologies and Tracking Dynamic Networks](https://mlsanity.com/paper/1610.08189/) (2016)
- [Disentangling Neurodegeneration with Brain Age Gap Prediction Models: A Graph Signal Processing Perspective](https://mlsanity.com/paper/2510.12763/) (2025)
- [LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees](https://mlsanity.com/paper/2305.01379/) (2023)
- [A Covariance Matching Approach to Graph Topology Identification](https://mlsanity.com/paper/2601.15999/) (2026)

## Abstract (arXiv)

We address the problem of inferring a directed network from nodal measurements generated by linear diffusion dynamics on the sought graph. Observations are modeled as the outputs of a graph convolutional filter, i.e., a polynomial (with unknown coefficients) of a local diffusion graph-shift operator encoding the latent graph topology, excited with an ensemble of independent graph signals with arbitrarily-correlated nodal components. Unlike prior efforts that considered undirected graphs and white signal excitations, here the graph-shift operator and the observations' covariance matrix are not simultaneously diagonalizable. In this challenging context, we first rely on measurements of the output signals along with prior statistical information on the inputs to identify the diffusion filter. Such system identification problem involves solving a system of quadratic matrix equations, which we show is identifiable under spectral-diversity assumptions on the input covariances. For algorithmic purposes we recast it as a smooth quadratic minimization subject to Stiefel manifold constraints. Subsequent identification of the network topology given the graph filter estimate boils down to finding a sparse and structurally admissible shift that commutes with the given filter, thus, forcing the latter to be a polynomial in the sought graph-shift operator. A joint graph filter and topology identification algorithm is also proposed, which alternates between the aforementioned steps in a mutually reinforcing fashion to offer improved sample complexity. Numerical tests corroborate the effectiveness of the proposed algorithms in recovering synthetic digraphs and real-data case studies, and illustrate their potential utility on urban mobility analyses as well as portfolio optimization.