stat.MLcs.LGcs.SIeess.SPTheoretical
Directed Graph Topology Inference via Graph Filter Identification
Jun 25, 2026
This paper addresses the problem of inferring a directed network from nodal measurements using graph convolutional filters and identifies the diffusion filter and network topology.
The paper considers the challenging context of inferring a directed network from nodal measurements with non-diagonalizable graph-shift operator and observations' covariance matrix.
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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.