The effective representation, precessing, analysis, and visualization of large-scale structured data over graphs are gaining a lot of attention. So far most of the literature has focused on real-valued signals. However, signals are often sparse in the Fourier domain, and more informative and compact representations for them can be obtained using the complex envelope of their spectral components, as opposed to the original real-valued signals. Motivated by this fact, in this work we generalize graph convolutional neural networks (GCN) to the complex domain, deriving the theory that allows to incorporate a complex-valued graph shift operators (GSO) in the definition of graph filters (GF) and process complex-valued graph signals (GS). The theory developed can handle spatio-temporal complex network processes. We prove that complex-valued GCNs are stable with respect to perturbations of the underlying graph support, the bound of the transfer error and the bound of error propagation through multiply layers. Then we apply complex GCN to power grid state forecasting, power grid cyber-attack detection and localization.
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