Luana Ruiz

Meher Chaitanya, My Le, Luana Ruiz

We introduce Graph Cascades, a mesoscopic rewiring strategy for Graph Neural Networks (GNNs) and Graph Transformers (GTs) that captures intermediate-scale graph structure beyond purely local edges or fully global attention. Using contagion-based diffusion processes, Graph Cascades constructs, in O(|V|+|E|) time, an auxiliary graph where node pairs supported by repeated multi-hop reinforcement are promoted to direct neighbors. We theoretically characterize when reinforcement-based rewiring helps: sufficient conditions under which reinforcement-based edge selection is more label-aligned than direct adjacency, an SBM witness in which two-hop reinforcement is perfectly homophilic, and a formalization of mesoscopic connectivity via graph effective resistance. Empirically, across node-classification benchmarks, Graph Cascades improves multiple GNN and sparse-GT backbones, with the most reliable gains observed on heterophilic and moderate- to high-degree homophilic graphs. The theoretical conditions also identify regimes where mesoscopic rewiring is unlikely to be beneficial -- low-degree regular graphs and graphs with structural bottlenecks -- and these predictions match the observed failures. We additionally observe tight correlations between performance and structural properties in the rewired graphs.

Juan Cervino, Luana Ruiz, Alejandro Ribeiro

Graph Neural Networks (GNNs) rely on graph convolutions to exploit meaningful patterns in networked data. Based on matrix multiplications, convolutions incur in high computational costs leading to scalability limitations in practice. To overcome these limitations, proposed methods rely on training GNNs in smaller number of nodes, and then transferring the GNN to larger graphs. Even though these methods are able to bound the difference between the output of the GNN with different number of nodes, they do not provide guarantees against the optimal GNN on the very large graph. In this paper, we propose to learn GNNs on very large graphs by leveraging the limit object of a sequence of growing graphs, the graphon. We propose to grow the size of the graph as we train, and we show that our proposed methodology -- learning by transference -- converges to a neighborhood of a first order stationary point on the graphon data. A numerical experiment validates our proposed approach.

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

Geometric deep learning has gained much attention in recent years due to more available data acquired from non-Euclidean domains. Some examples include point clouds for 3D models and wireless sensor networks in communications. Graphs are common models to connect these discrete data points and capture the underlying geometric structure. With the large amount of these geometric data, graphs with arbitrarily large size tend to converge to a limit model -- the manifold. Deep neural network architectures have been proved as a powerful technique to solve problems based on these data residing on the manifold. In this paper, we propose a manifold neural network (MNN) composed of a bank of manifold convolutional filters and point-wise nonlinearities. We define a manifold convolution operation which is consistent with the discrete graph convolution by discretizing in both space and time domains. To sum up, we focus on the manifold model as the limit of large graphs and construct MNNs, while we can still bring back graph neural networks by the discretization of MNNs. We carry out experiments based on point-cloud dataset to showcase the performance of our proposed MNNs.

Thien Le, Luana Ruiz, Stefanie Jegelka

Large-scale graph machine learning is challenging as the complexity of learning models scales with the graph size. Subsampling the graph is a viable alternative, but sampling on graphs is nontrivial as graphs are non-Euclidean. Existing graph sampling techniques require not only computing the spectra of large matrices but also repeating these computations when the graph changes, e.g., grows. In this paper, we introduce a signal sampling theory for a type of graph limit -- the graphon. We prove a Poincaré inequality for graphon signals and show that complements of node subsets satisfying this inequality are unique sampling sets for Paley-Wiener spaces of graphon signals. Exploiting connections with spectral clustering and Gaussian elimination, we prove that such sampling sets are consistent in the sense that unique sampling sets on a convergent graph sequence converge to unique sampling sets on the graphon. We then propose a related graphon signal sampling algorithm for large graphs, and demonstrate its good empirical performance on graph machine learning tasks.

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

The increasing availability of geometric data has motivated the need for information processing over non-Euclidean domains modeled as manifolds. The building block for information processing architectures with desirable theoretical properties such as invariance and stability is convolutional filtering. Manifold convolutional filters are defined from the manifold diffusion sequence, constructed by successive applications of the Laplace-Beltrami operator to manifold signals. However, the continuous manifold model can only be accessed by sampling discrete points and building an approximate graph model from the sampled manifold. Effective linear information processing on the manifold requires quantifying the error incurred when approximating manifold convolutions with graph convolutions. In this paper, we derive a non-asymptotic error bound for this approximation, showing that convolutional filtering on the sampled manifold converges to continuous manifold filtering. Our findings are further demonstrated empirically on a problem of navigation control.

Sanjukta Krishnagopal, Luana Ruiz

Graph neural networks (GNNs) achieve remarkable performance in graph machine learning tasks but can be hard to train on large-graph data, where their learning dynamics are not well understood. We investigate the training dynamics of large-graph GNNs using graph neural tangent kernels (GNTKs) and graphons. In the limit of large width, optimization of an overparametrized NN is equivalent to kernel regression on the NTK. Here, we investigate how the GNTK evolves as another independent dimension is varied: the graph size. We use graphons to define limit objects -- graphon NNs for GNNs, and graphon NTKs for GNTKs -- , and prove that, on a sequence of graphs, the GNTKs converge to the graphon NTK. We further prove that the spectrum of the GNTK, which is related to the directions of fastest learning which becomes relevant during early stopping, converges to the spectrum of the graphon NTK. This implies that in the large-graph limit, the GNTK fitted on a graph of moderate size can be used to solve the same task on the large graph, and to infer the learning dynamics of the large-graph GNN. These results are verified empirically on node regression and classification tasks.

Luana Ruiz, Ningyuan Huang, Soledad Villar

In this work we propose a random graph model that can produce graphs at different levels of sparsity. We analyze how sparsity affects the graph spectra, and thus the performance of graph neural networks (GNNs) in node classification on dense and sparse graphs. We compare GNNs with spectral methods known to provide consistent estimators for community detection on dense graphs, a closely related task. We show that GNNs can outperform spectral methods on sparse graphs, and illustrate these results with numerical examples on both synthetic and real graphs.

Yeganeh Alimohammadi, Luana Ruiz, Amin Saberi

We propose a theoretical framework for training Graph Neural Networks (GNNs) on large input graphs via training on small, fixed-size sampled subgraphs. This framework is applicable to a wide range of models, including popular sampling-based GNNs, such as GraphSAGE and FastGCN. Leveraging the theory of graph local limits, we prove that, under mild assumptions, parameters learned from training sampling-based GNNs on small samples of a large input graph are within an $ε$-neighborhood of the outcome of training the same architecture on the whole graph. We derive bounds on the number of samples, the size of the graph, and the training steps required as a function of $ε$. Our results give a novel theoretical understanding for using sampling in training GNNs. They also suggest that by training GNNs on small samples of the input graph, practitioners can identify and select the best models, hyperparameters, and sampling algorithms more efficiently. We empirically illustrate our results on a node classification task on large citation graphs, observing that sampling-based GNNs trained on local subgraphs 12$\times$ smaller than the original graph achieve comparable performance to those trained on the input graph.

Caio F. Deberaldini Netto, Zhiyang Wang, Luana Ruiz

Graph Neural Networks (GNNs) have gained popularity in various learning tasks, with successful applications in fields like molecular biology, transportation systems, and electrical grids. These fields naturally use graph data, benefiting from GNNs' message-passing framework. However, the potential of GNNs in more general data representations, especially in the image domain, remains underexplored. Leveraging the manifold hypothesis, which posits that high-dimensional data lies in a low-dimensional manifold, we explore GNNs' potential in this context. We construct an image manifold using variational autoencoders, then sample the manifold to generate graphs where each node is an image. This approach reduces data dimensionality while preserving geometric information. We then train a GNN to predict node labels corresponding to the image labels in the classification task, and leverage convergence of GNNs to manifold neural networks to analyze GNN generalization. Experiments on MNIST and CIFAR10 datasets demonstrate that GNNs generalize effectively to unseen graphs, achieving competitive accuracy in classification tasks.

Mika Sarkin Jain, Stefanie Jegelka, Ishani Karmarkar et al.

How can we subsample graph data so that a graph neural network (GNN) trained on the subsample achieves performance comparable to training on the full dataset? This question is of fundamental interest, as smaller datasets reduce labeling costs, storage requirements, and computational resources needed for training. Selecting an effective subset is challenging: a poorly chosen subsample can severely degrade model performance, and empirically testing multiple subsets for quality obviates the benefits of subsampling. Therefore, it is critical that subsampling comes with guarantees on model performance. In this work, we introduce new subsampling methods for graph datasets that leverage the Tree Mover's Distance to reduce both the number of graphs and the size of individual graphs. To our knowledge, our approach is the first that is supported by rigorous theoretical guarantees: we prove that training a GNN on the subsampled data results in a bounded increase in loss compared to training on the full dataset. Unlike existing methods, our approach is both model-agnostic, requiring minimal assumptions about the GNN architecture, and label-agnostic, eliminating the need to label the full training set. This enables subsampling early in the model development pipeline (before data annotation, model selection, and hyperparameter tuning) reducing costs and resources needed for storage, labeling, and training. We validate our theoretical results with experiments showing that our approach outperforms existing subsampling methods across multiple datasets.

Haolin Li, Haoyu Wang, Luana Ruiz

Graph Neural Networks (GNNs) excel in many graph machine learning tasks but face challenges when scaling to large networks. GNN transferability allows training on smaller graphs and applying the model to larger ones, but existing methods often rely on random subsampling, leading to disconnected subgraphs and reduced model expressivity. We propose a novel graph sampling algorithm that leverages feature homophily to preserve graph structure. By minimizing the trace of the data correlation matrix, our method better preserves the graph Laplacian trace -- a proxy for the graph connectivity -- than random sampling, while achieving lower complexity than spectral methods. Experiments on citation networks show improved performance in preserving Laplacian trace and GNN transferability compared to random sampling.

Roxanne Holden, Luana Ruiz

Graphons, as limits of graph sequences, provide a framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons yields operator-level convergence rates, enabling transferability analyses of GNNs. This note summarizes known bounds under no assumptions, global Lipschitz continuity, and piecewise-Lipschitz continuity, highlighting tradeoffs between assumptions and rates, and illustrating their empirical tightness on synthetic and real data.

Haoyu Wang, Renyuan Ma, Gonzalo Mateos et al.

We introduce a principled generative framework for graph signals that enables explicit control of feature heterophily, a key property underlying the effectiveness of graph learning methods. Our model combines a Lipschitz graphon-based random graph generator with Gaussian node features filtered through a smooth spectral function of the rescaled Laplacian. We establish new theoretical guarantees: (i) a concentration result for the empirical heterophily score; and (ii) almost-sure convergence of the feature heterophily measure to a deterministic functional of the graphon degree profile, based on a graphon-limit law for polynomial averages of Laplacian eigenvalues. These results elucidate how the interplay between the graphon and the filter governs the limiting level of feature heterophily, providing a tunable mechanism for data modeling and generation. We validate the theory through experiments demonstrating precise control of homophily across graph families and spectral filters.

Caio F. Deberaldini Netto, Zhiyang Wang, Luana Ruiz

We propose a graph semi-supervised learning framework for classification tasks on data manifolds. Motivated by the manifold hypothesis, we model data as points sampled from a low-dimensional manifold $\mathcal{M} \subset \mathbb{R}^F$. The manifold is approximated in an unsupervised manner using a variational autoencoder (VAE), where the trained encoder maps data to embeddings that represent their coordinates in $\mathbb{R}^F$. A geometric graph is constructed with Gaussian-weighted edges inversely proportional to distances in the embedding space, transforming the point classification problem into a semi-supervised node classification task on the graph. This task is solved using a graph neural network (GNN). Our main contribution is a theoretical analysis of the statistical generalization properties of this data-to-manifold-to-graph pipeline. We show that, under uniform sampling from $\mathcal{M}$, the generalization gap of the semi-supervised task diminishes with increasing graph size, up to the GNN training error. Leveraging a training procedure which resamples a slightly larger graph at regular intervals during training, we then show that the generalization gap can be reduced even further, vanishing asymptotically. Finally, we validate our findings with numerical experiments on image classification benchmarks, demonstrating the empirical effectiveness of our approach.

My Le, Luana Ruiz, Souvik Dhara

Learning node representations is a fundamental problem in graph machine learning. While existing embedding methods effectively preserve local similarity measures, they often fail to capture global functions like graph distances. Inspired by Bourgain's seminal work on Hilbert space embeddings of metric spaces (1985), we study the performance of local distance-preserving node embeddings. Known as landmark-based algorithms, these embeddings approximate pairwise distances by computing shortest paths from a small subset of reference nodes called landmarks. Our main theoretical contribution shows that random graphs, such as Erdos-Renyi random graphs, require lower dimensions in landmark-based embeddings compared to worst-case graphs. Empirically, we demonstrate that the GNN-based approximations for the distances to landmarks generalize well to larger real-world networks, offering a scalable and transferable alternative for graph representation learning.

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

This paper studies the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold, thus encoding geometric information. We consider convolutional MNNs and GNNs where the manifold and the graph convolutions are respectively defined in terms of the Laplace-Beltrami operator and the graph Laplacian. Using the appropriate kernels, we analyze both dense and moderately sparse graphs. We prove non-asymptotic error bounds showing that convolutional filters and neural networks on these graphs converge to convolutional filters and neural networks on the continuous manifold. As a byproduct of this analysis, we observe an important trade-off between the discriminability of graph filters and their ability to approximate the desired behavior of manifold filters. We then discuss how this trade-off is ameliorated in neural networks due to the frequency mixing property of nonlinearities. We further derive a transferability corollary for geometric graphs sampled from the same manifold. We validate our results numerically on a navigation control problem and a point cloud classification task.

Luana Ruiz, Luiz F. O. Chamon, Alejandro Ribeiro

Graph neural networks (GNNs) are composed of layers consisting of graph convolutions and pointwise nonlinearities. Due to their invariance and stability properties, GNNs are provably successful at learning representations from data supported on moderate-scale graphs. However, they are difficult to learn on large-scale graphs. In this paper, we study the problem of training GNNs on graphs of moderate size and transferring them to large-scale graphs. We use graph limits called graphons to define limit objects for graph filters and GNNs -- graphon filters and graphon neural networks (WNNs) -- which we interpret as generative models for graph filters and GNNs. We then show that graphon filters and WNNs can be approximated by graph filters and GNNs sampled from them on weighted and stochastic graphs. Because the error of these approximations can be upper bounded, by a triangle inequality argument we can further bound the error of transferring a graph filter or a GNN across graphs. Our results show that (i) the transference error decreases with the graph size, and (ii) that graph filters have a transferability-discriminability tradeoff that in GNNs is alleviated by the scattering behavior of the nonlinearity. These findings are demonstrated empirically in a movie recommendation problem and in a decentralized control task.

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

Graph Neural Networks (GNNs) show impressive performance in many practical scenarios, which can be largely attributed to their stability properties. Empirically, GNNs can scale well on large size graphs, but this is contradicted by the fact that existing stability bounds grow with the number of nodes. Graphs with well-defined limits can be seen as samples from manifolds. Hence, in this paper, we analyze the stability properties of convolutional neural networks on manifolds to understand the stability of GNNs on large graphs. Specifically, we focus on stability to relative perturbations of the Laplace-Beltrami operator. To start, we construct frequency ratio threshold filters which separate the infinite-dimensional spectrum of the Laplace-Beltrami operator. We then prove that manifold neural networks composed of these filters are stable to relative operator perturbations. As a product of this analysis, we observe that manifold neural networks exhibit a trade-off between stability and discriminability. Finally, we illustrate our results empirically in a wireless resource allocation scenario where the transmitter-receiver pairs are assumed to be sampled from a manifold.

Luana Ruiz, Joshua Ainslie, Santiago Ontañón

Deep learning models generalize well to in-distribution data but struggle to generalize compositionally, i.e., to combine a set of learned primitives to solve more complex tasks. In sequence-to-sequence (seq2seq) learning, transformers are often unable to predict correct outputs for longer examples than those seen at training. This paper introduces iterative decoding, an alternative to seq2seq that (i) improves transformer compositional generalization in the PCFG and Cartesian product datasets and (ii) evidences that, in these datasets, seq2seq transformers do not learn iterations that are not unrolled. In iterative decoding, training examples are broken down into a sequence of intermediate steps that the transformer learns iteratively. At inference time, the intermediate outputs are fed back to the transformer as intermediate inputs until an end-of-iteration token is predicted. We conclude by illustrating some limitations of iterative decoding in the CFQ dataset.

Juan Cervino, Luana Ruiz, Alejandro Ribeiro

Graph Neural Networks (GNN) rely on graph convolutions to learn features from network data. GNNs are stable to different types of perturbations of the underlying graph, a property that they inherit from graph filters. In this paper we leverage the stability property of GNNs as a typing point in order to seek for representations that are stable within a distribution. We propose a novel constrained learning approach by imposing a constraint on the stability condition of the GNN within a perturbation of choice. We showcase our framework in real world data, corroborating that we are able to obtain more stable representations while not compromising the overall accuracy of the predictor.

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

The paper defines and studies manifold (M) convolutional filters and neural networks (NNs). \emph{Manifold} filters and MNNs are defined in terms of the Laplace-Beltrami operator exponential and are such that \emph{graph} (G) filters and neural networks (NNs) are recovered as discrete approximations when the manifold is sampled. These filters admit a spectral representation which is a generalization of both the spectral representation of graph filters and the frequency response of standard convolutional filters in continuous time. The main technical contribution of the paper is to analyze the stability of manifold filters and MNNs to smooth deformations of the manifold. This analysis generalizes known stability properties of graph filters and GNNs and it is also a generalization of known stability properties of standard convolutional filters and neural networks in continuous time. The most important observation that follows from this analysis is that manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations. This is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks. The most important practical consequence of this analysis is to shed light on the behavior of graph filters and GNNs in large-scale graphs.

Juan Cervino, Luana Ruiz, Alejandro Ribeiro

Graph neural networks (GNNs) use graph convolutions to exploit network invariances and learn meaningful feature representations from network data. However, on large-scale graphs convolutions incur in high computational cost, leading to scalability limitations. Leveraging the graphon -- the limit object of a graph -- in this paper we consider the problem of learning a graphon neural network (WNN) -- the limit object of a GNN -- by training GNNs on graphs sampled from the graphon. Under smoothness conditions, we show that: (i) the expected distance between the learning steps on the GNN and on the WNN decreases asymptotically with the size of the graph, and (ii) when training on a sequence of growing graphs, gradient descent follows the learning direction of the WNN. Inspired by these results, we propose a novel algorithm to learn GNNs on large-scale graphs that, starting from a moderate number of nodes, successively increases the size of the graph during training. This algorithm is further benchmarked on a decentralized control problem, where it retains comparable performance to its large-scale counterpart at a reduced computational cost.

Luana Ruiz, Fernando Gama, Alejandro Ribeiro et al.

Graph convolutional neural networks (GCNNs) learn compositional representations from network data by nesting linear graph convolutions into nonlinearities. In this work, we approach GCNNs from a state-space perspective revealing that the graph convolutional module is a minimalistic linear state-space model, in which the state update matrix is the graph shift operator. We show that this state update may be problematic because it is nonparametric, and depending on the graph spectrum it may explode or vanish. Therefore, the GCNN has to trade its degrees of freedom between extracting features from data and handling these instabilities. To improve such trade-off, we propose a novel family of nodal aggregation rules that aggregate node features within a layer in a nonlinear state-space parametric fashion allowing for a better trade-off. We develop two architectures within this family inspired by the recurrence with and without nodal gating mechanisms. The proposed solutions generalize the GCNN and provide an additional handle to control the state update and learn from the data. Numerical results on source localization and authorship attribution show the superiority of the nonlinear state-space generalization models over the baseline GCNN.

Luana Ruiz, Zhiyang Wang, Alejandro Ribeiro

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.

Luana Ruiz, Fernando Gama, Alejandro Ribeiro

Graph Neural Networks (GNNs) are information processing architectures for signals supported on graphs. They are presented here as generalizations of convolutional neural networks (CNNs) in which individual layers contain banks of graph convolutional filters instead of banks of classical convolutional filters. Otherwise, GNNs operate as CNNs. Filters are composed with pointwise nonlinearities and stacked in layers. It is shown that GNN architectures exhibit equivariance to permutation and stability to graph deformations. These properties help explain the good performance of GNNs that can be observed empirically. It is also shown that if graphs converge to a limit object, a graphon, GNNs converge to a corresponding limit object, a graphon neural network. This convergence justifies the transferability of GNNs across networks with different number of nodes. Concepts are illustrated by the application of GNNs to recommendation systems, decentralized collaborative control, and wireless communication networks.

Luana Ruiz, Luiz F. O. Chamon, Alejandro Ribeiro

Graph neural networks (GNNs) rely on graph convolutions to extract local features from network data. These graph convolutions combine information from adjacent nodes using coefficients that are shared across all nodes. Since these coefficients are shared and do not depend on the graph, one can envision using the same coefficients to define a GNN on another graph. This motivates analyzing the transferability of GNNs across graphs. In this paper we introduce graphon NNs as limit objects of GNNs and prove a bound on the difference between the output of a GNN and its limit graphon-NN. This bound vanishes with growing number of nodes if the graph convolutional filters are bandlimited in the graph spectral domain. This result establishes a tradeoff between discriminability and transferability of GNNs.

Alejandro Parada-Mayorga, Luana Ruiz, Alejandro Ribeiro

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.

Luana Ruiz, Fernando Gama, Alejandro Ribeiro

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) as a general learning framework that achieves this goal by leveraging the notion of a recurrent hidden state 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 prove that GRNNs are permutation equivariant and that they are stable to perturbations of the underlying graph support. To address the problem of vanishing gradients, we also put forward gated GRNNs with three different gating mechanisms: time, node and edge gates. In numerical experiments involving both synthetic and real datasets, time-gated GRNNs are shown to improve upon GRNNs in problems with long term dependencies, while node and edge gates help encode long range dependencies present in the graph. The numerical results also show that GRNNs outperform GNNs and RNNs, highlighting the importance of taking both the temporal and graph structures of a graph process into account.

Luana Ruiz, Fernando Gama, Antonio G. Marques et al.

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.

Luana Ruiz, Fernando Gama, Alejandro Ribeiro

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.

Luana Ruiz, Fernando Gama, Antonio G. Marques et al.

Graph neural networks (GNNs) have been shown to replicate convolutional neural networks' (CNNs) superior performance in many problems involving graphs. By replacing regular convolutions with linear shift-invariant graph filters (LSI-GFs), GNNs take into account the (irregular) structure of the graph and provide meaningful representations of network data. However, LSI-GFs fail to encode local nonlinear graph signal behavior, and so do regular activation functions, which are nonlinear but pointwise. To address this issue, we propose median activation functions with support on graph neighborhoods instead of individual nodes. A GNN architecture with a trainable multirresolution version of this activation function is then tested on synthetic and real-word datasets, where we show that median activation functions can improve GNN capacity with marginal increase in complexity.