Network topology identification from imperfect spectral templates

This paper studies network topology inference, which is a cornerstone problem in statistical analysis of complex systems. The fresh look advocated here builds on recent advances in 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 the covariance 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, endows the network structure with certain desired properties such as sparsity. To that end we develop efficient inference algorithms stemming from provably-tight convex relaxations of natural non-convex criteria. We initially propose algorithms along with theoretical performance guarantees for the case when the eigenbasis is perfectly known. We then investigate setups where an imperfect (noisy) eigenbasis is available as well as others when only a subset of the eigenvectors is known. Numerical tests showcase the effectiveness of the proposed algorithms in recovering real-world social networks.