Network topology inference from non-stationary graph signals

We address the problem of inferring a graph from nodal observations, which are modeled as non-stationary graph signals generated by local diffusion dynamics that depend on the structure of the sought network. Using the so-called graph-shift operator (GSO) as a matrix representation of the graph, we first identify the eigenvectors of the shift matrix from realizations of the diffused signals, and then we rely on these spectral templates to estimate the eigenvalues by imposing desirable properties on the graph to be recovered. Different from the stationary setting where the GSO and the covariance matrix of the observed signals are simultaneously diagonalizable, here they are not. Hence, estimating the eigenvectors requires first estimating the unknown diffusion (graph) filter - a polynomial in the GSO which does preserve the sought eigenbasis. To carry out this initial system identification step, we leverage different sources of information on the input signal driving the diffusion process on the graph. Numerical tests showcase the effectiveness of the proposed algorithms in recovering social and structural brain graphs.