Statistical Graph Signal Processing: Stationarity and Spectral Estimation

Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many contemporary applications the information of interest resides in more irregular domains that can be conveniently represented using a graph. This chapter reviews recent advances in extending the notion of stationarity to random graph signals. This is a challenging task due to the irregularity of the underlying graph domain. To that end, we start by presenting coexisting stationarity definitions along with explanations of their genesis, advantages, and disadvantages. Second, we introduce the concept of power spectral density for graph processes and propose a number of methods for its estimation. These methods include nonparametric approaches such as correlograms and windowed average periodograms as well as parametric approaches. To account for distributed scenarios where the supporting graph is related to an actual network infrastructure, the last part of the chapter discusses how to estimate the power spectral density of a graph process when having access to only a subset of the nodes. To gain intuition and insights, the concepts and schemes presented throughout the chapter are illustrated with a running example based on a real-world social graph.