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.@en

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 practical scenarios the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. The contribution in this paper is twofold. Firstly, we propose a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical definition in the time domain. Provided that the topology of the graph can be described by a normal matrix, our definition models stationary graph processes as the output of a linear graph filter applied to a white input. We will show that this is equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform. Secondly, we analyze the properties of the power spectral density and propose a number of methods to estimate it. We start with nonparametric approaches, including periodograms, window-based average periodograms, and filter banks. We then shift the focus to parametric approaches, discussing the estimation of moving-average (MA), autoregressive (AR) and ARMA processes. Finally, we illustrate the power spectral density estimation in synthetic and real-world graphs signals.

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Advancing a holistic theory of networks and network processes requires the extension of existing results in the processing of time-varying signals to signals supported on graphs. This paper focuses on the definition of stationarity and power spectral density for random graph signals, generalizes the concepts of autoregressive and moving average random processes to the graph domain, and investigates their parametric spectral estimation. Theoretical and algorithmic results are complemented with numerical tests on synthetic and real-world graphs.

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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 practical scenarios the information of interest resides in more irregular graph domains. The contribution in this paper is twofold. First, we propose several equivalent notions of weak stationarity for random graph signals, all taking into account the structure of the graph where the random process takes place. Second, we analyze the properties of the induced power spectral density along with nonparametric approaches to estimate it, including average and window-based periodograms.@en

A novel scheme for sampling graph signals is proposed. Space-shift sampling can be understood as a hybrid scheme that combines selection sampling -- observing the signal values on a subset of nodes - and aggregation sampling - observing the signal values at a single node after successive aggregation of local data. Under the assumption of bandlimitedness, we state conditions and propose strategies for signal recovery in different settings. Being a more general procedure, space-shift sampling achieves smaller reconstruction errors than current schemes, as we illustrate through the reconstruction of the industrial activity in a graph of the U.S. economy.
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New schemes to recover signals defined in the nodes of a graph are proposed. Our focus is on reconstructing bandlimited graph signals, which are signals that admit a sparse representation in a frequency domain related to the structure of the graph. Most existing formulations focus on estimating an unknown graph signal by observing its value on a subset of nodes. By contrast, in this paper, we study the problem of inducing a known graph signal using as input a graph signal that is nonzero only for a small subset of nodes. The sparse signal is then percolated (interpolated) across the graph using a graph filter. Alternatively, one can interpret graph signals as network states and study graph-signal reconstruction as a network-control problem where the target class of states is represented by bandlimited signals. Three setups are investigated. In the first one, a single simultaneous injection takes place on several nodes in the graph. In the second one, successive value injections take place on a single node. The third one is a generalization where multiple nodes inject multiple signal values. For noiseless settings, conditions under which perfect reconstruction is feasible are given, and the corresponding schemes to recover the desired signal are specified. Scenarios leading to imperfect reconstruction, either due to insufficient or noisy signal value injections, are also analyzed. Moreover, connections with classical interpolation in the time domain are discussed. Specifically, for time-varying signals, where the ideal interpolator after uniform sampling is a (low-pass) filter, our proposed approach and the reconstruction of a sampled signal coincide. Nevertheless, for general graph signals, we show that these two approaches differ. The last part of the paper presents numerical experiments that illustrate the results developed through synthetic and real-world signal reconstruction problems.@en

A new scheme to sample signals defined on the nodes of a graph is proposed. The underlying assumption is that such signals admit a sparse representation in a frequency domain related to the structure of the graph, which is captured by the so-called graph-shift operator. Instead of using the value of the signal observed at a subset of nodes to recover the signal in the entire graph, the sampling scheme proposed here uses as input observations taken at a single node. The observations correspond to sequential applications of the graph-shift operator, which are linear combinations of the information gathered by the neighbors of the node. When the graph corresponds to a directed cycle (which is the support of time-varying signals), our method is equivalent to the classical sampling in the time domain. When the graph is more general, we show that the Vandermonde structure of the sampling matrix, critical when sampling time-varying signals, is preserved. Sampling and interpolation are analyzed first in the absence of noise, and then noise is considered. We then study the recovery of the sampled signal when the specific set of frequencies that is active is not known. Moreover, we present a more general sampling scheme, under which, either our aggregation approach or the alternative approach of sampling a graph signal by observing the value of the signal at a subset of nodes can be both viewed as particular cases. Numerical experiments illustrating the results in both synthetic and real-world graphs close the paper.@en