Network topology identification from spectral templates

Abstract

Network topology inference is a cornerstone problem in statistical analyses of complex systems. In this context, the fresh look advocated here permeates benefits from convex optimization and graph signal processing, to identify the so-termed graph shift operator (encoding the network topology) given only the eigenvectors of the shift. These spectral templates can be obtained, for example, from principal component analysis of a set of graph signals defined on the particular network. The novel idea is to find a graph shift that while being consistent with the provided spectral information, it endows the network structure with certain desired properties such as sparsity. The focus is on developing efficient recovery algorithms along with identifiability conditions for two particular shifts, the adjacency matrix and the normalized graph Laplacian. Application domains include network topology identification from steady-state signals generated by a diffusion process, and design of a graph filter that facilitates the distributed implementation of a prescribed linear network operator. Numerical tests showcase the effectiveness of the proposed algorithms in recovering synthetic and structural brain networks.

Learning sparse graphs under smoothness prior

“…The above problem (7) is noncovex due to the Boolean and cardinality constraints on w and the coupling between the optimization variables in the second term of (7). We provide two methods to solve it.…”

“…The optimization problem (7) can be solved using alternating minimization with respect to {x k } L k=1 and w. That is, given w, the problem in (7) reduces to a linear system in the unknown X, which admits a closed form solution; while given {x k } L k=1 , it reduces to a Boolean linear programming problem, which admits an analytical solution with respect to w based on rank ordering. These observations suggest an iterative alternating minimization algorithm yielding successive estimates of {x k } L k=1 with fixed w, and alternately of w with fixed {x k } L k=1 .…”

“…The iterations are initialized at i = 0 by randomly generating w[i + 1] from a uniform distribution over W. The above alternating minimization method is computationally very attractive, and consists of two simple known solutions per iteration. However, the algorithm converges only to a stationary point of (7), and it suffers from the choice of the initial estimate.…”

“…Graph topology identification is also investigated in [7] under the assumption that the eigenvectors of the graph Laplacian are known, which is a much stronger assumption. Although the eigenvectors can be computed from graph data (or the sample covariance matrix) when it is stationary with respect to the graph [8,9], the graph signals need not always be vertex stationary.…”

“…In any case, the estimated eigenvectors are not error free due to limited data records. In most of the existing approaches [4,5,7], graph sparsification is (or can be) achieved by penalizing the ℓ1-norm of the graph Laplacian matrix, adjacency matrix or the shift operator, however, there is no explicit handle to control the number of edges, unlike the proposed approach. In a related line of research, [10,11] investigate computing sparse graphs that approximate a given graph spectrally, which means that their Laplacian matrices have similar quadratic forms.…”