Directed Graph Topology Inference via Graph Filter Identification

2026-06-25

AI Summary

This study addresses the problem of inferring directed graph topologies from node observations generated by linear diffusion dynamics. The authors propose a novel method for jointly identifying the graph filter and the underlying graph topology: first, the filter is recovered by solving a smooth quadratic optimization problem under Stiefel manifold constraints; subsequently, a sparse graph shift operator that commutes with the identified filter is estimated. This approach is the first to achieve joint identification under non-white excitation and directed graph settings, thereby overcoming the conventional reliance on simultaneous diagonalizability of the graph operator and the observation covariance matrix. Experiments demonstrate that the method significantly improves sample efficiency and topological recovery accuracy on both synthetic directed graphs and real-world datasets, including urban mobility and portfolio data.

Abstract

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.

Problem

• directed graph
• topology inference
• graph filter identification
• network reconstruction
• graph signal processing

Innovation

• directed graph topology inference
• graph filter identification
• quadratic matrix equations
• Stiefel manifold optimization
• spectral diversity