Chapter 13 - Inference of Graph Topology

Abstract

Network topology inference is a prominent problem in network science. Most graph signal processing (GSP) efforts to date assume that the underlying network is known, and then analyze how the graph’s algebraic and spectral characteristics impact the properties of the graph signals of interest. However, such assumption is often untenable in practice and typically adopted graph construction schemes are largely informal, distinctly lacking an element of validation. The present chapter outlines a framework recently developed to bridge the aforementioned gap by using information available from graph signals to infer the underlying graph topology. The unknown graph represents direct relationships between signal elements, which one aims to recover from observable indirect relationships generated by a diffusion process on the graph. The fresh look advocated here leverages concepts 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 operator that, while being consistent with the provided spectral information, endows the network with certain desired properties such as sparsity or minimum-energy edge weights.