Machine learning and data processing techniques relying on covariance information are widespread as they identify meaningful patterns in unsupervised and unlabeled settings. As a prominent example, Principal Component Analysis (PCA) projects data points onto the eigenvectors of their covariance matrix, capturing the directions of maximum variance. This mapping, however, falls short in two directions: it fails to capture information in low-variance directions, relevant when, e.g., the data contains high-variance noise; and it provides unstable results in low-sample regimes, especially when covariance eigenvalues are close. CoVariance Neural Networks (VNNs), i.e., graph neural networks using the covariance matrix as a graph, show improved stability to estimation errors and learn more expressive functions in the covariance spectrum than PCA, but require training and operate in a labeled setup. To get the benefits of both worlds, we propose Covariance Scattering Transforms (CSTs), deep untrained networks that sequentially apply filters localized in the covariance spectrum to the input data and produce expressive hierarchical representations via nonlinearities. We define the filters as covariance wavelets that capture specific and detailed covariance spectral patterns. We improve CSTs' computational and memory efficiency via a pruning mechanism, and we prove that their error due to finite-sample covariance estimations is less sensitive to close covariance eigenvalues compared to PCA, improving their stability. Our experiments on age prediction from cortical thickness measurements on 4 datasets collecting patients with neurodegenerative diseases show that CSTs produce stable representations in low-data settings, as VNNs but without any training, and lead to comparable or better predictions w.r.t. more complex learning models.

In urban centers, cycling is increasingly popular as an eco-friendly transportation mode and a short-distance transport option, driving higher demand for accurate bicycle travel time estimation. Policymakers need to understand bicycle traffic for urban traffic management and sustainable transport promotion, while cyclists benefit from better route planning and improved network efficiency. However, urban bicycle travel time estimation has not received as much attention as car traffic estimation and presents several challenges: 1) Limited availability of structural cycling data, which can be inaccessible due to privacy concerns and/or severely biased by user demographics. 2) The diverse and complex behaviors of cyclists. 3) The lack of strict road constraints for cyclists and frequent rule violations, complicating the model definition of a comprehensive cycling infrastructure network. This paper presents the first study on urban bicycle travel time estimation using GPS tracking data. Leveraging graph-based deep learning’s ability to learn from topological network information, we introduce the Dual Graph-based approach for bicycles (DG4b), which employs two parallel encode-process-decode pipelines: one for a shared undirected road network graph to capture intrinsic road characteristics, and another for a directed trip-specific graph reflecting unique trip features. The outputs are combined to estimate road segment speeds and overall trip travel time. When applied to a real-world dataset from Berlin, our method shows superior accuracy and reliability compared to baseline models, while maintaining low complexity. Our approach provides a novel perspective on integrating bicycling-specific characteristics and aims to inspire more future research in bicycle-related traffic estimation.

The commodity and widespread use of online shopping are having an unprecedented impact on climate, with emission figures from key actors that are easily comparable to those of a large-scale metropolis. Despite online shopping being fueled by recommender systems (RecSys) algorithms, the role and potential of the latter in promoting more sustainable choices is little studied. One of the main reasons for this could be attributed to the lack of a dataset containing carbon footprint emissions for the items. While building such a dataset is a rather challenging task, its presence is pivotal for opening the doors to novel perspectives, evaluations, and methods for RecSys research. In this article, we target this bottleneck and study the environmental role of RecSys algorithms. First, we mine a dataset that includes carbon footprint emissions for its items. Then, we benchmark conventional RecSys algorithms in terms of accuracy and sustainability as two faces of the same coin. We find that RecSys algorithms optimized for accuracy overlook greenness and that longer recommendation lists are greener but less accurate. Then, we show that a simple reranking approach that accounts for the item’s carbon footprint can establish a better trade-off between accuracy and greenness. This reranking approach is modular, ready to use, and can be applied to any RecSys algorithm without the need to alter the underlying mechanisms or retrain models. Our results show that a small sacrifice of accuracy can lead to significant improvements of recommendation greenness across all algorithms and list lengths. Arguably, this accuracy-greenness trade-off could even be seen as an enhancement of user satisfaction, particularly for purpose-driven users who prioritize the environmental impact of their choices. We anticipate this work will serve as the starting point for studying RecSys for more sustainable recommendations.

Covariance-based data processing is widespread across signal processing and machine learning applications due to its ability to model data interconnectivities and dependencies. However, harmful biases in the data may become encoded in the sample covariance matrix and cause data-driven methods to treat different subpopulations unfairly. Existing works such as fair principal component analysis (PCA) mitigate these effects, but remain unstable in low sample regimes, which in turn may jeopardize the fairness goal. To address both biases and instability, we propose Fair coVariance Neural Networks (FVNNs), which perform graph convolutions on the covariance matrix for both fair and accurate predictions. Our FVNNs provide a flexible model compatible with several existing bias mitigation techniques. In particular, FVNNs allow for mitigating the bias in two ways: first, they operate on fair covariance estimates that remove biases from their principal components; second, they are trained in an end-to-end fashion via a fairness regularizer in the loss function so that the model parameters are tailored to solve the task directly in a fair manner. We prove that FVNNs are intrinsically fairer than analogous PCA approaches thanks to their stability in low sample regimes. We validate the robustness and fairness of our model on synthetic and real-world data, showcasing the flexibility of FVNNs along with the tradeoff between fair and accurate performance.

Graph filters are ubiquitous for processing data over graphs. However, most filters obtained from data are point-estimates and may be sensitive to changes in topology or data distributions. Thus, modeling uncertainty in filters is critical to quantify confidence in analyses or improve downstream tasks in low-data regimes. We introduce a Bayesian framework for graph filter design, termed Bayesian graph filters. Given input-output realizations on a graph, we obtain the posterior filter and prior filter precision hyper-parameters via a constrained EM algorithm. The posterior filter leads to uncertainty in its frequency response, which has implications for stability. We study the stability via the integral Lipschitz (IL) property and derive a lower bound for the probability of Bayesian filters being IL. Results show that Bayesian filters can be more stable across the spectrum and under perturbations, provide uncertainty estimates and can outperform point filters on multiple tasks.

Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing non-isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.

This paper introduces a probabilistic approach for tracking the dynamics of unweighted and directed graphs using state-space models (SSMs). Unlike conventional topology inference methods that assume static graphs and generate point-wise estimates, our method accounts for dynamic changes in the network structure over time. We model the network at each timestep as the state of the SSM, and use observations to update beliefs that quantify the probability of the network being in a particular state. Then, by considering the dynamics of transition and observation models through the update and prediction steps, respectively, the proposed method can incorporate the information of real-time graph signals into the beliefs. These beliefs provide a probability distribution of the network at each timestep, being able to provide both an estimate for the network and the uncertainty it entails. Our approach is evaluated through experiments with synthetic and real-world networks. The results demonstrate that our method effectively estimates network states and accounts for the uncertainty in the data, outperforming traditional techniques such as recursive least squares.

We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This approach is intended for learning flow-type data on networks where edge flows can be characterized by the discrete divergence and curl. Drawing upon the Hodge decomposition, we first develop classes of divergence-free and curl-free edge GPs, suitable for various applications. We then combine them to create Hodge-compositional edge GPs that are expressive enough to represent any edge function. These GPs facilitate direct and independent learning for the different Hodge components of edge functions, enabling us to capture their relevance during hyperparameter optimization. To highlight their practical potential, we apply them for flow data inference in currency exchange, ocean currents and water supply networks, comparing them to alternative models.

Graphs are widely used to represent complex information and signal domains with irregular support. Typically, the underlying graph topology is unknown and must be estimated from the available data. Common approaches assume pairwise node interactions and infer the graph topology based on this premise. In contrast, our novel method not only unveils the graph topology but also identifies three-node interactions, referred to in the literature as second-order simplicial complexes (SCs). We model signals using a graph autoregressive Volterra framework, enhancing it with structured graph Volterra kernels to learn SCs. We propose a mathematical formulation for graph and SC inference, solving it through convex optimization involving group norms and mask matrices. Experimental results on synthetic and real-world data showcase a superior performance for our approach compared to existing methods.

Temporal networks arise due to certain dynamics influencing their connections or due to the change in interactions between the nodes themselves, as seen for example in social networks. Such evolution can be algebraically represented by a three-way tensor, which lends itself to using tensor decompositions to study the underpinning factors driving the network evolution. Low rank tensor decompositions have been used for temporal networks but mostly with a focus on downstream tasks and have been seldom used to study the temporal network itself. Here, we use the tensor decomposition to identify a limited number of key mode graphs that can explain the temporal network, and which linear combination can represent its evolution. For this, we put for a novel graph-based tensor decomposition approach where we impose a graph structure on the two modes of the tensor and a smoothness on the temporal dimension. We use these mode graphs to investigate the temporal network and corroborate their usability for network reconstruction and link prediction.

Simplicial complexes prove effective in modeling data with multi-way dependencies, such as data defined along the edges of networks or within other higher-order structures. Their spectrum can be decomposed into three interpretable subspaces via the Hodge decomposition, resulting foundational in numerous applications. We leverage this decomposition to develop a contrastive self-supervised learning approach for processing simplicial data and generating embeddings that encapsulate specific spectral information. Specifically, we encode the pertinent data invariances through simplicial neural networks and devise augmentations that yield positive contrastive examples with suitable spectral properties for downstream tasks. Additionally, we reweight the significance of negative examples in the contrastive loss, considering the similarity of their Hodge components to the anchor. By encouraging a stronger separation among less similar instances, we obtain an embedding space that reflects the spectral properties of the data. The numerical results on two standard edge flow classification tasks show a superior performance even when compared to supervised learning techniques. Our findings underscore the importance of adopting a spectral perspective for contrastive learning with higher-order data.

This paper proposes a novel algorithm to retroactively compute the evolution of edge signals from a given sequence of partial observations from topological structures, a concept referred to as evolution backcasting. Our backcasting algorithm exploits the spatio-temporal dependencies present in the real-world edge signals using the simplicial vector autoregressive (S-VAR) model. The proposed algorithm jointly estimates the S-VAR filter coefficients and recovers missing data from the partial observations. Subsequently, the algorithm capitalizes on the learned S-VAR model and the reconstructed signals to execute the backcasting of edge signal evolution. Using traffic and water distribution networks as case studies, we showcase the superior capabilities of our algorithm compared with baseline alternatives.

Inference of time varying data over graphs is of importance in real-world applications such as urban water networks, economics, and brain recordings. It typically relies on identifying a computationally affordable joint spatiotemporal method that can leverage the patterns in the data. While this per se is a challenging task, it becomes even more so when the network comes with uncertainties, which, if not accounted for, can lead to unpredictable consequences. To target this setting, we model graph uncertainties as Gaussian noise on the edges and design a stochastic partial differential equation (SPDE) based on it. We use this SPDE as a state equation to model the time varying signal evolution and extend it further to a state-space model where the observations are graph-filtered versions of the state. This allows us to have a joint spatiotemporal expressive kernel that can be estimated online via Kalman filtering and which parameters can also be estimated online via maximum likelihood principles, ultimately, reducing the computational cost. We corroborate the proposed approach on numerical experiments, showing a superior performance to approaches ignoring either the uncertainty or considering a separable spatiotemporal kernel.

Current spatiotemporal learning methods for complex data exploit the graph structure as an inductive bias to restrict the function space and improve data and computation efficiency. However, these methods work principally on graphs with a fixed size, whereas in several applications there are expanding graphs where new nodes join the network; e.g., new sensors joining a sensor network or new users joining a recommender system. This paper focuses on the non-trivial extension of spatiotemporal methods to this setting, where now it is key to jointly capture both the topological and signal dynamics. Specifically, it considers a graph vector autoregressive (GVAR) model for multivariate time series. The GVAR is a multivariate linear model that leverages a bank of graph filters allowing scalability and data efficiency. To account for the dynamic nature of the graphs, the filters’s parameters are learned on-the-fly via adaptive gradient descent with provable sub-linear regret. Numerical results on both synthetic and real data corroborate the proposed method.

Vector autoregressive (VAR) model is widely used to model time-varying processes, but it suffers from prohibitive growth of the parameters when the number of time series exceeds a few hundreds. We propose a simplicial VAR model to mitigate the curse of dimensionality of the VAR models when the time series are defined over higher-order network structures such as edges, triangles, etc. The proposed model shares parameters across the simplicial signals by leveraging the simplicial convolutional filter and captures structure-aware spatio-temporal dependencies of the time-varying processes. Targetting the streaming signals from the real-world nonstationary networks, we develop a group-lasso-based online strategy to learn the proposed model. Using traffic and water distribution networks, we demonstrate that the proposed model achieves competitive signal prediction accuracy with a significantly less number of parameters than the VAR models.

Simplicial convolutional filters can process signals defined over levels of a simplicial complex such as nodes, edges, triangles, and so on with applications in e.g., flow prediction in transportation or financial networks. However, the underlying topology expands over time in a way that new edges and triangles form. For example, in a transportation network, a new connection between two locations is newly built, or in a currency exchange market, two currencies can be exchanged without an intermediate currency that can be understood as a new edge between them. To handle the streaming nature of data, we propose an online prediction for edge flows which generalizes to other higher-order simplicial signals. This is achieved by updating the filter coefficients via an online gradient descent with a provable sub-linear regret relative to the simplicial filter optimized over the whole sequence of edge flows. The update of the filter coefficients associated with the lower and upper Hodge Laplacians can be uncoupled in general. We test the online edge flow prediction on an expanding synthetic simplicial complex and a coauthorship complex showing a close performance to the offline counterpart.

Devising and analysing learning models for spatiotemporal network data is of importance for tasks including forecasting, anomaly detection, and multi-agent coordination, among others. Graph Convolutional Neural Networks (GCNNs) are an established approach to learn from time-invariant network data. The graph convolution operation offers a principled approach to aggregate information and offers mathematical analysis by exploring tools from graph signal processing. This analysis provides insights into the equivariance properties of GCNNs; spectral behaviour of the learned filters; and the stability to graph perturbations, which arise from support perturbations or uncertainties. However, extending the convolutional learning and respective analysis to the spatiotemporal domain is challenging because spatiotemporal data have more intrinsic dependencies. Hence, a higher flexibility to capture jointly the spatial and temporal dependencies is required to learn meaningful higher-order representations. Here, we leverage product graphs to represent the spatiotemporal dependencies in the data and introduce Graph-Time Convolutional Neural Networks (GTCNNs) as a principled architecture. We also introduce a parametric product graph to learn the spatiotemporal coupling. The convolution principle further allows a similar mathematical tractability as for GCNNs. In particular, the stability result shows GTCNNs are stable to spatial perturbations. owever, there is an implicit trade-off between discriminability and robustness; i.e., the more complex the model, the less stable. Extensive numerical results on benchmark datasets corroborate our findings and show the GTCNN compares favorably with state-of-the-art solutions. We anticipate the GTCNN to be a starting point for more sophisticated models that achieve good performance but are also fundamentally grounded.

An online algorithm for missing data imputation for networks with signals defined on the edges is presented. Leveraging the prior knowledge intrinsic to real-world networks, we propose a bi-level optimization scheme that exploits the causal dependencies and the flow conservation, respectively via(i)a sparse line graph identification strategy based on a group-Lasso and(ii)a Kalman filtering-based signal reconstruction strategy developed using simplicial complex (SC) formulation. The advantages of this first SC-based attempt for time-varying signal imputation have been demonstrated through numerical experiments using EPANET models of both synthetic and real water distribution networks.

Performing signal processing over graphs requires knowledge of the underlying fixed topology. However, graphs often grow in size with new nodes appearing over time, whose connectivity is typically unknown; hence, making more challenging the downstream tasks in applications like cold start recommendation. We address such a challenge for signal interpolation at the incoming nodes blind to the topological connectivity of the specific node. Specifically, we propose a stochastic attachment model for incoming nodes parameterized by the attachment probabilities and edge weights. We estimate these parameters in a data-driven fashion by relying only on the attachment behaviour of earlier incoming nodes with the goal of interpolating the signal value. We study the non-convexity of the problem at hand, derive conditions when it can be marginally convexified, and propose an alternating projected descent approach between estimating the attachment probabilities and the edge weights. Numerical experiments with synthetic and real data dealing in cold start collaborative filtering corroborate our findings.

This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network structures. These filters are built by following the shift-and-sum principle of the convolution operation and rely on the Hodge-Laplacians to shift the signal within the simplex. But since in SCs we have also inter-simplex coupling, we use the incidence matrices to transfer the signal in adjacent simplices and build a filter bank to jointly filter signals from different levels. We prove some interesting properties for the proposed filter bank, including permutation and orientation equivariance, a computational complexity that is linear in the SC dimension, and a spectral interpretation using the simplicial Fourier transform. We illustrate the proposed approach with numerical experiments.

Graphs can model networked data by representing them as nodes and their pairwise relationships as edges. Recently, signal processing and neural networks have been extended to process and learn from data on graphs, with achievements in tasks like graph signal reconstruction, graph or node classifications, and link prediction. However, these methods are only suitable for data defined on the nodes of a graph. In this paper, we propose a simplicial convolutional neural network (SCNN) architecture to learn from data defined on simplices, e.g., nodes, edges, triangles, etc. We study the SCNN permutation and orientation equivariance, complexity, and spectral analysis. Finally, we test the SCNN performance for imputing citations on a coauthorship complex.

Driven by the outstanding performance of neural networks in the structured euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. For instance, we show that GCNNs have a parameter sharing structure that induces permutation equivariance. This can be an advantage or a limitation, depending on the application. In cases where it is a limitation, we propose hybrid approaches and provide insights to develop several other solutions that promote parameter sharing without enforcing permutation equivariance. Another interesting conclusion is the unification of GCNNs and GATs —approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power.

Stochastic graph neural networks (SGNNs) are information processing architectures that can learn representations from data over random graphs. SGNNs are trained with respect to the expected performance, but this training comes with no guarantee about the deviation of particular output realizations around the optimal mean. To overcome this issue, we propose a learning strategy for SGNNs based on a variance constrained optimization problem, balancing the expected performance and the stochastic deviation. To handle the variance constraint in the stochastic optimization problem, training is undertaken in the dual domain. We propose an alternating primal-dual learning algorithm that updates the primal variable (SGNN parameters) with gradient descent and the dual variable with gradient ascent. We show the stochastic deviation is explicitly controlled through Chebyshev inequality and analyze the optimality loss induced by the primal-dual learning. Through numerical simulations, we observe a strong performance in expectation with a controllable deviation corroborating the theoretical findings.

To deal with high-dimensional data, graph filters have shown their power in both graph signal processing and data science. However, graph filters process signals exploiting only pairwise interactions between the nodes, and they are not able to exploit more complicated topological structures. Graph Volterra models, on the other hand, are also able to exploit relations between triplets, quadruplets and so on. However, they have only been exploited for topology identification and are only based on one-hop relations. In this paper, we first review graph filters and graph Volterra models and then merge the two concepts resulting in so-called topological Volterra filters (TVFs). TVFs process signals over multiple hops of higher-level topological structures. First-level TVFs are basically similar to traditional graph filters, yet higher-level TVFs provide a more general processing framework. We apply TVFs to inverse filtering and recommender systems.

Signal processing and machine learning algorithms for data sup-ported over graphs, require the knowledge of the graph topology. Unless this information is given by the physics of the problem (e.g., water supply networks, power grids), the topology has to be learned from data. Topology identification is a challenging task, as the problem is often ill-posed, and becomes even harder when the graph structure is time-varying. In this paper, we address the problem of dynamic topology identification by building on recent results from time-varying optimization, devising a general-purpose online algorithm operating in non-stationary environments. Because of its iteration-constrained nature, the proposed approach exhibits an intrinsic temporal-regularization of the graph topology without explicitly enforcing it. As a case-study, we specialize our method to the Gaussian graphical model (GGM) problem and corroborate its performance.

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.

Graph neural networks (GNNs) have recently gained significant momentum in the recommendation community, demonstrating state-of-the-art performance in top-k recommendation and next-item recommendation. Despite promising results on GNN-based recommendation and search, most of the current GNN research remains essentially concentrated on more traditional tasks such as classification or regression. The GReS workshop on Graph Neural Networks for Recommendation and Search is then a first endeavor to bridge the gap between the RecSys and GNN communities, and promote recommendation and search problems amongst GNN practitioners.

Graph convolutions, in both their linear and neural network forms, have reached state-of-the-art accuracy on recommender system (RecSys) benchmarks. However, recommendation accuracy is tied with diversity in a delicate trade-off and the potential of graph convolutions to improve the latter is unexplored. Here, we develop a model that learns joint convolutional representations from a nearest neighbor and a furthest neighbor graph to establish a novel accuracy-diversity trade-off for recommender systems. The nearest neighbor graph connects entities (users or items) based on their similarities and is responsible for improving accuracy, while the furthest neighbor graph connects entities based on their dissimilarities and is responsible for diversifying recommendations. The information between the two convolutional modules is balanced already in the training phase through a regularizer inspired by multi-kernel learning. We evaluate the joint convolutional model on three benchmark datasets with different degrees of sparsity. The proposed method can either trade accuracy to improve substantially the catalog coverage or the diversity within the list; or improve both by a lesser amount. Compared with accuracy-oriented graph convolutional approaches, the proposed model shows diversity gains up to seven times by trading as little as 1% in accuracy. Compared with alternative accuracy-diversity trade-off solutions, the joint graph convolutional model retains the highest accuracy while offering a handle to increase diversity. To our knowledge, this is the first work proposing an accuracy-diversity trade-off with graph convolutions and opens the doors to learning over graphs approaches for improving such trade-off.

Diffusion-based semi-supervised learning on graphs consists of diffusing labeled information of a few nodes to infer the labels on the remaining ones. The performance of these methods heavily relies on the initial labeled set, which is either generated randomly or using heuristics. The first sometimes leads to unsatisfactory results because random labeling has no guarantees to label all classes while heuristic methods only yield a good performance when multiple recursive training stages are possible. In this paper, we put forth a new paradigm for one-shot active semi-supervised learning for graph diffusions. We rephrase active learning as the problem of selecting the output labels from a label propagation model. Subsequently, we develop two methods to solve this problem and label the nodes. The first method assumes there are only a few starting labels and relies on projected compressive sensing to build the label set. The second method drops the assumption of a few starting labels and builds on sparse sensing techniques to label a few nodes. Both methods have solid mathematical grounds in signal processing and require a single training phase. Numerical results on three scenarios corroborate our findings and showcase the improved performance compared with the state of the art.

Graph neural networks (GNNs) model nonlinear representations in graph data with applications in distributed agent coordination, control, and planning among others. However, current GNN implementations assume ideal distributed scenarios and ignore link fluctuations that occur due to environment or human factors. In these situations, the GNN fails to address its distributed task if the topological randomness is not considered accordingly. To overcome this issue, we put forth the stochastic graph neural network (SGNN) model: a GNN where the distributed graph convolutional operator is modified to account for the network changes. Since stochasticity brings in a new paradigm, we develop a novel learning process for the SGNN and introduce the stochastic gradient descent (SGD) algorithm to estimate the parameters. We prove through the SGD that the SGNN learning process converges to a stationary point under mild Lipschitz assumptions. Numerical simulations corroborate the proposed theory and show an improved performance of the SGNN compared with the conventional GNN when operating over random time varying graphs.

The forecasting of multi-variate time processes through graph-based techniques has recently been addressed under the graph signal processing framework. However, problems in the representation and the processing arise when each time series carries a vector of quantities rather than a scalar one. To tackle this issue, we devise a new framework and propose new methodologies based on the graph vector autoregressive model. More explicitly, we leverage product graphs to model the high-dimensional graph data and develop multidimensional graph-based vector autoregressive models to forecast future trends with a number of parameters that is independent of the number of time series and a linear computational complexity. Numerical results demonstrating the prediction of moving point clouds corroborate our findings.

A critical challenge in graph signal processing is the sampling of bandlimited graph signals; signals that are sparse in a well-defined graph Fourier domain. Current works focused on sampling time-invariant graph signals and ignored their temporal evolution. However, time can bring new insights on sampling since sensor, biological, and financial network signals are correlated in both domains. Hence, in this work, we develop a sampling theory for time varying graph signals, named graph processes, to observe and track a process described by a linear state-space model. We provide a mathematical analysis to highlight the role of the graph, process bandwidth, and sample locations. We also propose sampling strategies that exploit the coupling between the topology and the corresponding process. Numerical experiments corroborate our theory and show the proposed methods trade well the number of samples with accuracy.

In this work, we jointly exploit tools from graph signal processing and control theory to drive a bandlimited graph signal that is being diffused on a random time-varying graph from a subset of nodes. As our main contribution, we rely only on the statistics of the graph to introduce the concept of controllability in the mean, and therefore drive the signal on the expected graph to a desired bandlimited state. A mean-square error (MSE) analysis is performed for two main tasks: i) to highlight the role played by the signal bandwidth and the control nodes to the deviation from the mean signal of a particular realization; and ii) to select the control nodes and design the control signal that minimize this MSE. Numerical results validate the introduced controllability in the mean framework and show its ability to cope with time-varying topologies.

This letter investigates methods to detect graph topological changes without making any assumption on the nature of the change itself. To accomplish this, we merge recently developed tools in graph signal processing with matched subspace detection theory and propose two blind topology change detectors. The first detector exploits the prior information that the observed signal is sparse w.r.t. the graph Fourier transform of the nominal graph, while the second makes use of the smoothness prior w.r.t. the nominal graph to detect topological changes. Both detectors are compared with their respective nonblind counterparts in a synthetic scenario that mimics brain networks. The absence of information about the alternative graph, in some cases, might heavily influence the blind detector's performance. However, in cases where the observed signal deviates slightly from the nonblind model, the information about the alternative graph turns out to be not useful.

Generally in distributed signal processing, and specifically in distributed graph filters, reducing the communication and computational complexity plays a key role in the network lifetime. In this work we present a novel algorithm to sparsify the graph filtering operation in a random way, where each node decides locally with a certain probability with which of its neighbors to communicate. We show that, if the filter coefficients are changed accordingly, the first and second order moment of the stochastic output are identical to the deterministic filter output and bounded, respectively. We apply our idea on the tasks of signal denoising and diffusion. Numerical results show that the distributed implementation costs of the filter can be reduced up to 95% with a variance of 10-3from the deterministic output.

We present a novel implementation strategy for distributed autoregressive moving average (ARMA) graph filters. Differently from the state of the art implementation, the proposed approach has the following benefits: (i) the designed filter coefficients come with stability guarantees, (ii) the linear convergence time can now be controlled by the filter coefficients, and (iii) the stable filter coefficients that approximate a desired frequency response are optimal in a least squares sense. Numerical results show that the proposed implementation outperforms the state of the art distributed infinite impulse response (IIR) graph filters. Further, even at fixed distributed costs, compared with the popular finite impulse response (FIR) filters, at high orders our method achieves tighter low-pass responses, suggesting that it should be preferable in accuracy-demanding applications.