Researchers Crack Directed Graph Inference from Diffusion Measurements

Inferring the structure of a directed network from observed node-level signals is a key challenge in fields from neuroscience to finance. A new paper by Shafipour, Buciulea, Segarra, Marques, and Mateos tackles this problem for data generated by linear diffusion dynamics—modeled as a graph convolutional filter with unknown coefficients. Previous methods relied on simplifying assumptions: undirected graphs and white (uncorrelated) signal excitations. This work lifts both restrictions, handling arbitrary correlations among input signals and directed edges (so the graph-shift operator and covariance matrix are not simultaneously diagonalizable). The authors first identify the diffusion filter by solving a system of quadratic matrix equations, reformulated as a smooth optimization on a Stiefel manifold. They prove identifiability under spectral-diversity conditions.

Once the filter is known, the topology is recovered by finding a sparse, structurally admissible shift matrix that commutes with it. The paper also introduces a joint algorithm that alternates between filter and topology identification, improving sample complexity. Numerical experiments on synthetic digraphs and real-world data—including urban mobility analyses and portfolio optimization—validate the method's effectiveness. Submitted to IEEE Transactions on Signal Processing, this work provides a rigorous framework for network discovery in complex systems where directionality and correlated inputs are the norm.