Graphons are infinite-dimensional objects that represent the limit of convergent sequences of discrete graphs. This paper derives a theory of Graphon Signal Processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters. These two objects are graphon counterparts of graph Fourier transforms and graph filters. It is shown that in convergent sequences of graphs and associated graph signals: (i) The graph Fourier transform converges to the graphon Fourier transform when considering graphon bandlimited signals. (ii) The spectral and vertex responses of graph filters converge to the spectral and vertex responses of graphon filters with the same coefficients. These theorems imply that for graphs that belong to certain families -in the sense that they are part of sequences that converge to a certain graphon-graph Fourier analysis and graph filter design have well defined limits. In turn, these facts extend applicability of graph signal processing to graphs with large number of nodes -because we can transfer designs from limit graphons to finite graphs-and dynamic graphs -because we can transfer designs to different graphs drawn from the same graphon.

Graph processes exhibit a temporal structure determined by the sequence index and and a spatial structure determined by the graph support. To learn from graph processes, an information processing architecture must then be able to exploit both underlying structures. We introduce Graph Recurrent Neural Networks (GRNNs), which achieve this goal by leveraging the hidden Markov model (HMM) together with graph signal processing (GSP). In the GRNN, the number of learnable parameters is independent of the length of the sequence and of the size of the graph, guaranteeing scalability. We also prove that GRNNs are permutation equivariant and that they are stable to perturbations of the underlying graph support. Following the observation that stability decreases with longer sequences, we propose a time-gated extension of GRNNs. We also put forward node-and edge-gated variants of the GRNN to address the problem of vanishing gradients arising from long range graph dependencies. The advantages of GRNNs over GNNs and RNNs are demonstrated in a synthetic regression experiment and in a classification problem where seismic wave readings from a network of seismographs are used to predict the region of an earthquake. Finally, the benefits of time, node and edge gating are experimentally validated in multiple time and spatial correlation scenarios.

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Graph signals are signals with an irregular structure that can be described by a graph. Graph neural networks (GNNs) are information processing architectures tailored to these graph signals and made of stacked layers that compose graph convolutional filters with nonlinear activation functions. Graph convolutions endow GNNs with invariance to permutations of the graph nodes' labels. In this paper, we consider the design of trainable nonlinear activation functions that take into consideration the structure of the graph. This is accomplished by using graph median filters and graph max filters, which mimic linear graph convolutions and are shown to retain the permutation invariance of GNNs. We also discuss modifications to the backpropagation algorithm necessary to train local activation functions. The advantages of localized activation function architectures are demonstrated in four numerical experiments: source localization on synthetic graphs, authorship attribution of 19th century novels, movie recommender systems and scientific article classification. In all cases, localized activation functions are shown to improve model capacity.

In many network problems, graphs may change by the addition of nodes, or the same problem may need to be solved in multiple similar graphs. This generates inefficiency, as analyses and systems that are not transferable have to be redesigned. To address this, we consider graphons, which are both limit objects of convergent graph sequences and random graph models. We define graphon signals and introduce the Graphon Fourier Transform (WFT), to which the Graph Fourier Transform (GFT) is shown to converge. This result is demonstrated in two numerical experiments where, as expected, the GFT converges, hinting to the possibility of centralizing analysis and design on graphons to leverage transferability.

Graph processes model a number of important problems such as identifying the epicenter of an earthquake or predicting weather. In this paper, we propose a Graph Convolutional Recurrent Neural Network (GCRNN) architecture specifically tailored to deal with these problems. GCRNNs use convolutional filter banks to keep the number of trainable parameters independent of the size of the graph and of the time sequences considered. We also put forward Gated GCRNNs, a time-gated variation of GCRNNs akin to LSTMs. When compared with GNNs and another graph recurrent architecture in experiments using both synthetic and real-word data, GCRNNs significantly improve performance while using considerably less parameters.

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Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. Relying on the use of shift-invariant graph filters, GNNs extend the operation of convolution to graphs. However, the operations of pooling and sampling are still not clearly defined and the approaches proposed in the literature either modify the graph structure in a way that does not preserve its spectral properties, or require defining a policy for selecting which nodes to keep. In this work, we propose a new strategy for pooling and sampling on GNNs using graphons which preserves the spectral properties of the graph. To do so, we consider the graph layers in a GNN as elements of a sequence of graphs that converge to a graphon. In this way we have no ambiguity in the node labeling when mapping signals from one layer to the other and a spectral representation that is consistent throughout the layers. We evaluate this strategy in a synthetic and a real-world numerical experiment where we show that graphon pooling GNNs are less prone to overfitting and improve upon other pooling techniques, especially when the dimensionality reduction ratios between layers is large.