Network Topology Inference from Spectral Templates

Abstract

Abstract-We address the problem of identifying a graph structure from the observation of signals defined on its nodes. Fundamentally, the unknown graph encodes direct relationships between signal elements, which we aim to recover from observable indirect relationships generated by a diffusion process on the graph. The fresh look advocated here permeates benefits from convex optimization and stationarity of graph signals, in order to identify the graph shift operator (a matrix representation of the graph) given only its eigenvectors. These spectral templates can be obtained, e.g., from the sample covariance of independent graph signals diffused on the sought network. The novel idea is to find a graph shift that, while being consistent with the provided spectral information, endows the network with certain desired properties such as sparsity. To that end we develop efficient inference algorithms stemming from provably-tight convex relaxations of natural nonconvex criteria, particularizing the results for two shifts: the adjacency matrix and the normalized Laplacian. Algorithms and theoretical recovery conditions are developed not only when the templates are perfectly known, but also when the eigenvectors are noisy or when only a subset of them are given. Numerical tests showcase the effectiveness of the proposed algorithms in recovering social, brain, and amino-acid networks.

State-Space Network Topology Identification From Partial Observations

“…where diag(·) denotes a diagonal matrix with its argument on the main diagonal and S is the set of desired graph matrices, e.g., adjacency matrices, combinatorial Laplacian matrices, etc. To do so, we can employ methods existing in the GSP literature that, given the graph matrix eigenbasis, retrieve a sparse matrix representation of the graph [18,35].…”

“…However, in other cases, the network information is unknown and needs to be estimated. As the importance of studying such structures in the data has been noticed, retrieving the topology of the network has become a topic of extensive research [16][17][18][19][20][21][22][23][24].Despite the extensive research done so far (for a comprehensive review the reader is referred to [2,17] and references therein), most of the approaches do not lever a physical model beyond the one induced by the so-called graph filters [18,25] drawn from graph signal processing (GSP) [10,26,27]. Among the ones that propose a different interaction model e.g., [20,24], none of them considers the network data (a.k.a.…”

Network Inference From Consensus Dynamics With Unknown Parameters

“…where we fixed a desired probability level at 1−δ. Our goal now is to choose l small enough to ensure that (18) is satisfied and then solve for the corresponding number of observations M δ in (21) using such an l. To do this, first recall that 1 = h k (λ 1 ) > · · · > h k (λ N ). We further assume that h k (λ i ) > h k (λ j ) + τ when i < j for some τ > 0, where τ does not depend on M .…”

Blind Inference of Centrality Rankings from Graph Signals

“…Network topology inference has been studied from a statistical perspective where each node is a random variable, and edges reflect the covariance structure of the ensemble of random variables [10][11][12][13][14]. Additionally, graph signal processing methods have arisen recently, which infer network topology by assuming the observed signals are the output of some underlying network process [15][16][17][18][19]. Our work differs from this line of research, in that we do not infer the network structure, but rather the centrality ranking of the nodes.…”