Abstract
We address the problem of inferring an undirected graph from nodal observations, which are modeled as non-stationary graph signals generated by local diffusion dynamics that depend on the structure of the unknown network. Using the so-called graph-shift operator (GSO), which is a matrix representation of the graph, we first identify the eigenvectors of the shift matrix from observations of the diffused signals, and then estimate the eigenvalues by imposing desirable properties on the graph to be recovered. Different from the stationary setting where the eigenvectors can be obtained directly from the covariance matrix of the measurements, here we need to estimate first the unknown diffusion (graph) filter - a polynomial in the GSO that preserves the sought eigenbasis. To carry out this initial system identification step, we exploit different sources of information on the arbitrarily-correlated input signal driving the diffusion on the graph. We first explore the setting where the observations, the input information, and the unknown graph filter are linearly related. We then address the case where the relation is given by a system of matrix quadratic equations, which arises in pragmatic scenarios where only the second-order statistics of the inputs are available. While such a quadratic filter identification problem boils down to a non-convex fourth-order polynomial minimization, we discuss identifiability conditions, propose algorithms to approximate the solution, and analyze their performance. Numerical tests illustrate the effectiveness of the proposed topology inference algorithms in recovering brain, social, financial, and urban transportation networks using synthetic and real-world signals.
Highlights
C ONSIDER a network represented as a weighted and undirected graph G, consisting of a node set N of cardinality N, an edge set E of unordered pairs of elements in N, and edge weights Aij ∈ R such that Aij = Aji = 0 for all (i, j) ∈ E
Problem 1 Given a set Y := {y(p)}Pp=1 of P i.i.d. samples of a graph non-stationary random signal y adhering to the network diffusion model (2) possibly contaminated by additive noise, identify the sparse graph-shift operator (GSO) S assuming: a) knowledge of the respective input signals {x(p)}Pp=1; or b) sparsity of the input signals; or c) that the inputs signals are drawn from a zero-mean distribution with known covariance matrix
Given realizations of M diffusion processes {ym}M m=1 arranged as columns of matrix Y ∈ RN×M, a possible blind formulation of the graph filter identification problem amounts to finding H ∈ HN such that Y − HX is small for some matrix norm of interest, where the unobserved matrix X = [x1, ..., xM ] is assumed to be sparse
Summary
Works have recently explored this approach by identifying a GSO given its eigenvectors [12], [28], or given an estimate of the invertible diffusion kernel that generates the Gaussian observations [33], but all rely on measurements of stationary graph signals. Relative to its conference precursors [1], [2], in this journal paper we consider identification of undirected graphs from observations of diffused non-stationary processes in various settings through a unified presentation along with full-blown technical details (including extended discussions and unpublished proofs in the Appendices for all the theoretical results). Ker(X) refers to the null space of X and the spectral radius of matrix X is denoted by λmax(X)
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