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Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon perturbations. We then extend this analysis by interpreting the graphon neural network as a generating model for GNNs on deterministic and stochastic graphs instantiated from the original and perturbed graphons. We observe that GNNs are stable to graphon perturbations with a stability bound that decreases asymptotically with the size of the graph. This asymptotic behavior is further demonstrated in an experiment of movie recommendation.
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In this work, we propose to train a graph neural network via resampling from a graphon estimate obtained from the underlying network data. More specifically, the graphon or the link probability matrix of the underlying network is first obtained from which a new network will be resampled and used during the training process at each layer. Due to the uncertainty induced from the resampling, it helps mitigate the well-known issue of over-smoothing in a graph neural network (GNN) model. Our framework is general, computationally efficient, and conceptually simple. Another appealing feature of our method is that it requires minimal additional tuning during the training process. Extensive numerical results show that our approach is competitive with and in many cases outperform the other over-smoothing reducing GNN training methods.
Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. A key property of GCNNs is their stability to graph perturbations. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to stochastic graph perturbations induced by link losses. In particular, it proves the expected output difference between the GCNN over random perturbed graphs and the GCNN over the nominal graph is upper bounded by a factor that is linear in the link loss probability. We perform the stability analysis in the graph spectral domain such that the result holds uniformly for any graph. This result also shows the role of the nonlinearity and the architecture width and depth, and allows identifying handle to improve the GCNN robustness. Numerical simulations on source localization and robot swarm control corroborate our theoretical findings.
Graph neural networks (GNNs) are naturally distributed architectures for learning representations from network data. This renders them suitable candidates for decentralized tasks. In these scenarios, the underlying graph often changes with time due to link failures or topology variations, creating a mismatch between the graphs on which GNNs were trained and the ones on which they are tested. Online learning can be leveraged to retrain GNNs at testing time to overcome this issue. However, most online algorithms are centralized and usually offer guarantees only on convex problems, which GNNs rarely lead to. This paper develops the Wide and Deep GNN (WD-GNN), a novel architecture that can be updated with distributed online learning mechanisms. The WD-GNN consists of two components: the wide part is a linear graph filter and the deep part is a nonlinear GNN. At training time, the joint wide and deep architecture learns nonlinear representations from data. At testing time, the wide, linear part is retrained, while the deep, nonlinear one remains fixed. This often leads to a convex formulation. We further propose a distributed online learning algorithm that can be implemented in a decentralized setting. We also show the stability of the WD-GNN to changes of the underlying graph and analyze the convergence of the proposed online learning procedure. Experiments on movie recommendation, source localization and robot swarm control corroborate theoretical findings and show the potential of the WD-GNN for distributed online learning.
Our digital world is full of time series and graphs which capture the various aspects of many complex systems. Traditionally, there are respective methods in processing these two different types of data, e.g., Recurrent Neural Network (RNN) and Graph Neural Network (GNN), while in recent years, time series could be mapped to graphs by using the techniques such as Visibility Graph (VG), so that researchers can use graph algorithms to mine the knowledge in time series. Such mapping methods establish a bridge between time series and graphs, and have high potential to facilitate the analysis of various real-world time series. However, the VG method and its variants are just based on fixed rules and thus lack of flexibility, largely limiting their application in reality. In this paper, we propose an Adaptive Visibility Graph (AVG) algorithm that can adaptively map time series into graphs, based on which we further establish an end-to-end classification framework AVGNet, by utilizing GNN model DiffPool as the classifier. We then adopt AVGNet for radio signal modulation classification which is an important task in the field of wireless communication. The simulations validate that AVGNet outperforms a series of advanced deep learning methods, achieving the state-of-the-art performance in this task.
Graph neural networks (GNNs) have been successfully employed in a myriad of applications involving graph-structured data. Theoretical findings establish that GNNs use nonlinear activation functions to create low-eigenvalue frequency content that can be processed in a stable manner by subsequent graph convolutional filters. However, the exact shape of the frequency content created by nonlinear functions is not known, and thus, it cannot be learned nor controlled. In this work, node-variant graph filters (NVGFs) are shown to be capable of creating frequency content and are thus used in lieu of nonlinear activation functions. This results in a novel GNN architecture that, although linear, is capable of creating frequency content as well. Furthermore, this new frequency content can be either designed or learned from data. In this way, the role of frequency creation is separated from the nonlinear nature of traditional GNNs. Extensive simulations are carried out to differentiate the contributions of frequency creation from those of the nonlinearity.
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