In wireless networks characterized by dense connectivity, the significant signaling overhead generated by distributed link scheduling algorithms can exacerbate issues like congestion, energy consumption, and radio footprint expansion. To mitigate these challenges, we propose a distributed link sparsification scheme employing graph neural networks (GNNs) to reduce scheduling overhead for delay-tolerant traffic while maintaining network capacity. A GNN module is trained to adjust contention thresholds for individual links based on traffic statistics and network topology, enabling links to withdraw from scheduling contention when they are unlikely to succeed. Our approach is facilitated by a novel offline constrained unsupervised learning algorithm capable of balancing two competing objectives: minimizing scheduling overhead while ensuring that total utility meets the required level. In simulated wireless multi-hop networks with up to 500 links, our link sparsification technique effectively alleviates network congestion and reduces radio footprints across four distinct distributed link scheduling protocols.

Graphons are continuous models that represent the structure of graphs and allow the generation of graphs of varying sizes. We propose Scalable Implicit Graphon Learning (SIGL), a scalable method that combines implicit neural representations (INRs) and graph neural networks (GNNs) to estimate a graphon from observed graphs. Unlike existing methods, which face important limitations like fixed resolution and scalability issues, SIGL learns a continuous graphon at arbitrary resolutions. GNNs are used to determine the correct node ordering, improving graph alignment. Furthermore, we characterize the asymptotic consistency of our estimator, showing that more expressive INRs and GNNs lead to consistent estimators. We evaluate SIGL in synthetic and real-world graphs, showing that it outperforms existing methods and scales effectively to larger graphs, making it ideal for tasks like graph data augmentation.

A significant challenge for computation offloading in wireless multi-hop networks is the complex interactions among traffic flows in the presence of interference. Existing approaches often ignore these key effects and/or rely on outdated queueing and channel state information. To fill these gaps, we reformulate joint offloading and routing as a routing problem on an extended graph with physical and virtual links. We adopt the state-of-the-art shortest path-biased Backpressure routing algorithm, which allows the destination and the route of a job to be dynamically adjusted at every time step based on network-wide long-term information and real-time states of local neighborhoods. In large networks, our approach achieves smaller makespan than existing approaches, such as separated Backpressure offloading, and joint offloading and routing based on linear programming.

Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.

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.

This work presents a low-rank tensor model for multidimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.

Graph neural networks (GNNs) have become a workhorse approach for learning from data defined over irregular domains, typically by implicitly assuming that the data structure is represented by a homophilic graph. However, recent works have revealed that many relevant applications involve heterophilic data where the performance of GNNs can be notably compromised. To address this challenge, we present a simple yet effective architecture designed to mitigate the limitations of the homophily assumption. The proposed architecture reinterprets the role of graph filters in convolutional GNNs, resulting in a more general architecture while incorporating a stronger inductive bias than GNNs based on filter banks. The proposed convolutional layer enhances the expressive capacity of the architecture enabling it to learn from both homophilic and heterophilic data and preventing the issue of oversmoothing. From a theoretical standpoint, we show that the proposed architecture is permutation equivariant. Finally, we show that the proposed GNNs compares favorably relative to several state-of-the-art baselines in both homophilic and heterophilic datasets, showcasing its promising potential.

Score Distillation Sampling (SDS) has been pivotal for leveraging pre-trained diffusion models in downstream tasks such as inverse problems, but it faces two major challenges: $(i)$ mode collapse and $(ii)$ latent space inversion, which become more pronounced in high-dimensional data. To address mode collapse, we introduce a novel variational framework for posterior sampling. Utilizing the Wasserstein gradient flow interpretation of SDS, we propose a multimodal variational approximation with a \emph{repulsion} mechanism that promotes diversity among particles by penalizing pairwise kernel-based similarity. This repulsion acts as a simple regularizer, encouraging a more diverse set of solutions. To mitigate latent space ambiguity, we extend this framework with an \emph{augmented} variational distribution that disentangles the latent and data. This repulsive augmented formulation balances computational efficiency, quality, and diversity. Extensive experiments on linear and nonlinear inverse tasks with high-resolution images ($512 \times 512$) using pre-trained Stable Diffusion models demonstrate the effectiveness of our approach.

This paper proposes an efficient approach to joint task offloading and routing for real-time sensor data analytics at the network edge, enabling applications such as video surveillance and environmental monitoring. This problem can be formulated as a mixed-integer program (MIP) with the objective of utility maximization subject to the constraints of network topology, limited link capacity, and diverse task profiles. To efficiently approximate this NP-hard problem, we propose SeLR, a combination of primal-dual optimization and reweighted L1-norm regularization, which iteratively solves the convex relaxation while penalizing constraint violations and encouraging sparsity. Compared to greedy heuristics, SeLR provides a better accuracy—latency trade-off and better scalability to larger problems. Moreover, it reduces scheduling runtime by up to 9.17× over optimal solvers in networks with 300 nodes and 100 tasks.

Graphons, as limit objects of dense graph sequences, play a central role in the statistical analysis of network data. However, existing graphon estimation methods often struggle with scalability to large networks and resolution-independent approximation, due to their reliance on estimating latent variables or costly metrics such as the Gromov-Wasserstein distance. In this work, we propose a novel, scalable graphon estimator that directly recovers the graphon via moment matching, leveraging implicit neural representations (INRs). Our approach avoids latent variable modeling by training an INR--mapping coordinates to graphon values--to match empirical subgraph counts (i.e., moments) from observed graphs. This direct estimation mechanism yields a polynomial-time solution and crucially sidesteps the combinatorial complexity of Gromov-Wasserstein optimization. Building on foundational results, we establish a theoretical guarantee: when the observed subgraph motifs sufficiently represent those of the true graphon (a condition met with sufficiently large or numerous graph samples), the estimated graphon achieves a provable upper bound in cut distance from the ground truth. Additionally, we introduce MomentMixup, a data augmentation technique that performs mixup in the moment space to enhance graphon-based learning. Our graphon estimation method achieves strong empirical performance--demonstrating high accuracy on small graphs and superior computational efficiency on large graphs--outperforming state-of-the-art scalable estimators in 75\% of benchmark settings and matching them in the remaining cases. Furthermore, MomentMixup demonstrated improved graph classification accuracy on the majority of our benchmarks.

State-space models are a popular statistical framework for analysing sequential data. Within this framework, particle filters are often used to perform inference on non-linear state-space models. We introduce a new method, StateMixNN, that uses a pair of neural networks to learn the proposal distribution and transition kernel of a particle filter. Both distributions are approximated using multivariate Gaussian mixtures. The component means and covariances of these mixtures are learnt as outputs of learned functions. Our method is trained targeting the log-likelihood, thereby requiring only the observation series, and combines the interpretability of state-space models with the flexibility and approximation power of artificial neural networks . The proposed method significantly improves recovery of the hidden state in comparison with the state-of-the-art, showing greater improvement in highly non-linear scenarios.

With the increasing popularity of graph-based methods for dimensionality reduction and representation learning, node embedding functions have become important objects of study in the literature. In this paper, we take an axiomatic approach to understanding node embedding methods. Motivated by desirable properties of node embeddings for encoding the role of a node in the structure of a network, we first state three properties for embedding dissimilarity networks. We then prove that no node embedding method can satisfy all three properties at once, reflecting fundamental difficulties inherent to the task. Having identified these difficulties, we show that mild relaxations of these axioms allow for certain node embedding methods to be admissible.

We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on a maximum likelihood or maximum a posteriori approach using (simple) priors on the precision matrix, we consider a prior on the graph and rely on annealed Langevin diffusion to generate samples from the posterior distribution. Since the Langevin sampler requires access to the score function of the underlying graph prior, we use graph neural networks to effectively estimate the score from a graph dataset (either available beforehand or generated from a known distribution). Numerical experiments in different setups demonstrate the benefits of our approach.

Many problems that cloud operators solve are computationally expensive, and operators often use heuristic algorithms (that are faster and scale better than optimal) to solve them more efficiently. Heuristic analyzers enable operators to find when and by how much their heuristics underperform. However, these tools do not provide enough detail for operators to mitigate the heuristic's impact in practice: they only discover a single input instance that causes the heuristic to underperform (and not the full set) and they do not explain why.

We propose a performance analysis tool for learning-enabled systems that allows operators to uncover potential performance issues before deploying DNNs in their systems. The tools that exist for this purpose require operators to faithfully model all components (a white-box approach) or do inefficient black-box local search. We propose a gray-box alternative, which eliminates the need to precisely model all the system's components. Our approach is faster and finds substantially worse scenarios compared to prior work. We show that a state-of-the-art learning-enabled traffic engineering pipeline can underperform the optimal by 6× --- a much higher number compared to what the authors found.

Computational offloading has become an enabling component for edge intelligence in mobile and smart devices. Existing offloading schemes mainly focus on mobile devices and servers, while ignoring the potential network congestion caused by tasks from multiple mobile devices, especially in wireless multi-hop networks. To fill this gap, we propose a low-overhead, congestion-aware distributed task offloading scheme by augmenting a distributed greedy framework with graph-based machine learning. In simulated wireless multi-hop networks with 20-110 nodes and a resource allocation scheme based on shortest path routing and contention-based link scheduling, our approach is demonstrated to be effective in reducing congestion or unstable queues under the context-agnostic baseline, while improving the execution latency over local computing.

We introduce a new method, named PropMixNN, that uses a neural network to learn the proposal distribution of a particle filter. The optimal proposal distribution is approximated as a multivariate Gaussian mixture, so the proposed method aims at learning the means and covariance matrices of the S components that characterise the mixture. This unsupervised method is trained to target the log-likelihood, which does not require knowledge of the hidden state. The performance of the method is assessed in a stochastic Lorenz 96 model, which presents a non-linear chaotic behaviour. The proposed method reduces estimation errors in comparison with the state-of-the-art, showing greater improvement in highly non-linear scenarios.

In this work, we propose data augmentation via pairwise mixup across subgroups to improve group fairness. Many real-world applications of machine learning systems exhibit biases across certain groups due to underrepresentation or training data that reflects societal biases. Inspired by the successes of mixup for improving classification performance, we develop a pairwise mixup scheme to augment training data and encourage fair and accurate decision boundaries for all subgroups. Data augmentation for group fairness allows us to add new samples of underrepresented groups to balance subpopulations. Furthermore, our method allows us to use the generalization ability of mixup to improve both fairness and accuracy. We compare our proposed mixup to existing data augmentation and bias mitigation approaches on both synthetic simulations and real-world benchmark fair classification data, demonstrating that we are able to achieve fair outcomes with robust if not improved accuracy.

The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.

We propose a sampling algorithm to perform system identification from a set of input-output graph signal pairs. The dynamics of the systems we study are given by a partially known adjacency matrix and a generic parametric graph filter of unknown parameters. The methodology we employ is built upon the principles of annealed Langevin diffusion. This enables us to draw samples from the posterior distribution instead of following the classical approach of point estimation using maximum likelihood. We investigate how to harness the prior information inherent in a dataset of graphs of different sizes through the utilization of graph neural networks. We demonstrate, via numerical experiments involving both real-world and synthetic networks, that integrating prior knowledge into the estimation process enhances estimation performance.

Node features bolster graph-based learning when exploited jointly with network structure. However, a lack of nodal attributes is prevalent in graph data. We present a framework to recover completely missing node features for a set of graphs, where we only know the signals of a subset of graphs. Our approach incorporates prior information from both graph topology and existing nodal values. We demonstrate an example implementation of our framework where we assume that node features depend on local graph structure. Missing nodal values are estimated by aggregating known features from the most similar nodes. Similarity is measured through a node embedding space that preserves local topological features, which we train using a Graph AutoEncoder. We empirically show not only the accuracy of our feature estimation approach but also its value for downstream graph classification. Our success embarks on and implies the need to emphasize the relationship between node features and graph structure in graph-based learning.

We propose a joint channel estimation and data detection algorithm for massive multilple-input multiple-output systems based on diffusion models. Our proposed method solves the blind inverse problem by sampling from the joint posterior distribution of the symbols and channels and computing an approximate maximum a posteriori estimation. To achieve this, we construct a diffusion process that models the joint distribution of the channels and symbols given noisy observations, and then run the reverse process to generate the samples. A unique contribution of the algorithm is to include the discrete prior distribution of the symbols and a learned prior for the channels. Indeed, this is key as it allows a more efficient exploration of the joint search space and, therefore, enhances the sampling process. Through numerical experiments, we demonstrate that our method yields a lower normalized mean squared error than competing approaches and reduces the pilot overhead.

We propose estimating Gaussian graphical models (GGMs) that are fair with respect to sensitive nodal attributes. Many real-world models exhibit unfair discriminatory behavior due to biases in data. Such discrimination is known to be exacerbated when data is equipped with pairwise relationships encoded in a graph. Additionally, the effect of biased data on graphical models is largely underexplored. We thus introduce fairness for graphical models in the form of two bias metrics to promote balance in statistical similarities across nodal groups with different sensitive attributes. Leveraging these metrics, we present Fair GLASSO, a regularized graphical lasso approach to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across groups. We also propose an efficient proximal gradient algorithm to obtain the estimates. Theoretically, we express the tradeoff between fair and accurate estimated precision matrices. Critically, this includes demonstrating when accuracy can be preserved in the presence of a fairness regularizer. On top of this, we study the complexity of Fair GLASSO and demonstrate that our algorithm enjoys a fast convergence rate. Our empirical validation includes synthetic and real-world simulations that illustrate the value and effectiveness of our proposed optimization problem and iterative algorithm.

MOTIVATION: The study of bacterial genome dynamics is vital for understanding the mechanisms underlying microbial adaptation, growth, and their impact on host phenotype. Structural variants (SVs), genomic alterations of 50 base pairs or more, play a pivotal role in driving evolutionary processes and maintaining genomic heterogeneity within bacterial populations. While SV detection in isolate genomes is relatively straightforward, metagenomes present broader challenges due to the absence of clear reference genomes and the presence of mixed strains. In response, our proposed method rhea, forgoes reference genomes and metagenome-assembled genomes (MAGs) by encompassing all metagenomic samples in a series (time or other metric) into a single co-assembly graph. The log fold change in graph coverage between successive samples is then calculated to call SVs that are thriving or declining. RESULTS: We show rhea to outperform existing methods for SV and horizontal gene transfer (HGT) detection in two simulated mock metagenomes, particularly as the simulated reads diverge from reference genomes and an increase in strain diversity is incorporated. We additionally demonstrate use cases for rhea on series metagenomic data of environmental and fermented food microbiomes to detect specific sequence alterations between successive time and temperature samples, suggesting host advantage. Our approach leverages previous work in assembly graph structural and coverage patterns to provide versatility in studying SVs across diverse and poorly characterized microbial communities for more comprehensive insights into microbial gene flux. AVAILABILITY AND IMPLEMENTATION: rhea is open source and available at: https://github.com/treangenlab/rhea.

We develop an efficient and near-optimal solution for beamforming in multi-user multiple-input-multiple-output single-hop wireless ad-hoc interference networks. Inspired by the weighted minimum mean squared error (WMMSE) method, a classical approach to solving this problem, and the principle of algorithm unfolding, we present unfolded WMMSE (UWMMSE) for MU-MIMO. This method learns a parameterized functional transformation of key WMMSE variables using graph neural networks (GNNs), where the channel and interference components of a wireless network constitute the underlying graph. These GNNs are trained through gradient descent on a network utility metric using multiple instances of the beamforming problem. Comprehensive experimental analyses illustrate the superiority of UWMMSE over the classical WMMSE and state-of-the-art learning-based methods in terms of performance, generalizability, and robustness.

We propose a novel data-driven approach to allocate transmit power for federated learning (FL) over interference-limited wireless networks. The proposed method is useful in challenging scenarios where the wireless channel is changing during the FL training process and when the training data are not independent and identically distributed (non-i.i.d.) on the local devices. Intuitively, the power policy is designed to optimize the information received at the server end during the FL process under communication constraints. Ultimately, our goal is to improve the accuracy and efficiency of the global FL model being trained. The proposed power allocation policy is parameterized using graph convolutional networks (GCNs), and the associated constrained optimization problem is solved through a primal-dual (PD) algorithm. Theoretically, we show that the formulated problem has a zero duality gap and, once the power policy is parameterized, optimality depends on how expressive this parameterization is. Numerically, we demonstrate that the proposed method outperforms existing baselines under different wireless channel settings and varying degrees of data heterogeneity.

We leverage a novel telemetry source available in public clouds today: periodic summaries of every flow that enters or leaves any VM. A key aspect is that such telemetry can be collected transparently to customers and with minimal impact on their workloads. By consuming this telemetry, we show how one may realize complete and dynamic graphs of the communication inside cloud subscriptions. We describe novel analyses over these communication graphs with implications on network security and management.

Analyzing network topologies and communication graphs is essential in modern network management. However, the lack of a cohesive approach results in a steep learning curve, increased errors, and inefficiencies. In this paper, we present a novel approach that enables natural-language-based network management experiences, leveraging large language models (LLMs) to generate task-specific code from natural language queries. This method addresses the challenges of explainability, scalability, and privacy by allowing network operators to inspect the generated code, removing the need to share network data with LLMs, and focusing on application-specific requests combined with program synthesis techniques. We develop and evaluate a prototype system using benchmark applications, demonstrating high accuracy, cost-effectiveness, and potential for further improvements using complementary program synthesis techniques.

Stroke is one of the leading causes of death worldwide, and its mortality rate is drastically higher for patients who suffer recurrent strokes. Motivated by the recent success of graph learning methods on medical tasks, we introduce a graph representation framework for stroke recurrence prediction (GraSReP) based on patient data. In a nutshell, GraSReP sequentially consists of: i) a procedure for converting tabular, time-series patient data to a series of graphs, ii) a graph deep learning architecture that provides embeddings for the patients and, iii) a random forest classifier that predicts the patients’ risk of recurrent stroke based on these graph embeddings. We demonstrate GraSReP’s effectiveness for predicting recurrent strokes using real-world electronic health records, and discuss how it can be leveraged for the efficient application of preventive care.

We develop a novel data-driven nonlinear mixup mechanism for graph data augmentation and present different mixup functions for sample pairs and their labels. Mixup is a data augmentation method to create new training data by linearly interpolating between pairs of data samples and their labels. Mixup of graph data is challenging since the interpolation between graphs of potentially different sizes is an ill-posed operation. Hence, a promising approach for graph mixup is to first project the graphs onto a common latent feature space and then explore linear and nonlinear mixup strategies in this latent space. In this context, we propose to (i) project graphs onto the latent space of continuous random graph models known as graphons, (ii) leverage convex clustering in this latent space to generate nonlinear data-driven mixup functions, and (iii) investigate the use of different mixup functions for labels and data samples. We evaluate our graph data augmentation performance on benchmark datasets and demonstrate that nonlinear data-driven mixup functions can significantly improve graph classification.

We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our framework leverages the structure of the eigenmodes of the Hodge Laplacian of the product space to jointly filter along time and space. To this end, we provide a decomposition theorem of the Hodge Laplacian of the product space, which highlights how the product structure induces a decomposition of each eigenmode into a spatial and temporal component. Finally, we apply our method to real world data, specifically for interpolating trajectories of buoys in the ocean from a limited set of observed trajectories.

We consider the task of representing signals supported on graph bundles, which are generalizations of product graphs that allow for "twists" in the product structure. Leveraging the localized product structure of a graph bundle, we demonstrate how a suitable partition of unity over the base graph can be used to lift the signal on the graph into a space where a product factorization can be readily applied. Motivated by the locality of this procedure, we demonstrate that bases for the signal spaces of the components of the graph bundle can be lifted in the same way, yielding a basis for the signal space of the total graph. We demonstrate this construction on synthetic graphs, as well as with an analysis of the energy landscape of conformational manifolds in stereochemistry.

We propose a throughput-optimal biased backpressure (BP) algorithm for routing, where the bias is learned through a graph neural network that seeks to minimize end-to-end delay. Classical BP routing provides a simple yet powerful distributed solution for resource allocation in wireless multi-hop networks but has poor delay performance. A low-cost approach to improve this delay performance is to favor shorter paths by incorporating pre-defined biases in the BP computation, such as a bias based on the shortest path (hop) distance to the destination. In this work, we improve upon the widely-used metric of hop distance (and its variants) for the shortest path bias by introducing a bias based on the link duty cycle, which we predict using a graph convolutional neural network. Numerical results show that our approach can improve the delay performance compared to classical BP and existing BP alternatives based on pre-defined bias while being adaptive to interference density. In terms of complexity, our distributed implementation only introduces a one-time overhead (linear in the number of devices in the network) compared to classical BP, and a constant overhead compared to the lowest-complexity existing bias-based BP algorithms.

We propose a multiple-input multiple-output (MIMO) detector based on an annealed version of the underdamped Langevin (stochastic) dynamic. Our detector achieves state-of-the-art performance in terms of symbol error rate (SER) while keeping the computational complexity in check. Indeed, our method can be easily tuned to strike the right balance between computational complexity and performance as required by the application at hand. This balance is achieved by tuning hyperparameters that control the length of the simulated Langevin dynamic. Through numerical experiments, we demonstrate that our detector yields lower SER than competing approaches (including learning-based ones) with a lower running time compared to a previously proposed overdamped Langevin-based MIMO detector.

MOTIVATION: Model organisms are widely used to better understand the molecular causes of human disease. While sequence similarity greatly aids this cross-species transfer, sequence similarity does not imply functional similarity, and thus, several current approaches incorporate protein-protein interactions to help map findings between species. Existing transfer methods either formulate the alignment problem as a matching problem which pits network features against known orthology, or more recently, as a joint embedding problem. RESULTS: We propose a novel state-of-the-art joint embedding solution: Embeddings to Network Alignment (ETNA). ETNA generates individual network embeddings based on network topological structure and then uses a Natural Language Processing-inspired cross-training approach to align the two embeddings using sequence-based orthologs. The final embedding preserves both within and between species gene functional relationships, and we demonstrate that it captures both pairwise and group functional relevance. In addition, ETNA's embeddings can be used to transfer genetic interactions across species and identify phenotypic alignments, laying the groundwork for potential opportunities for drug repurposing and translational studies. AVAILABILITY AND IMPLEMENTATION: https://github.com/ylaboratory/ETNA.

We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable $[0,1]$-valued functions on unit cubes of increasing dimension, each corresponding to a dimension of the abstract simplicial complex. We show that convergence in homomorphism density implies convergence in a cut-metric, and vice versa, as well as showing that simplicial complexes sampled from the limit objects closely resemble its structure. Applying this framework, we also partially characterize the convergence of nonuniform hypergraphs.

A widely established set of unsupervised node embedding methods can be interpreted as consisting of two distinctive steps: i) the definition of a similarity matrix based on the graph of interest followed by ii) an explicit or implicit factorization of such matrix. Inspired by this viewpoint, we propose improvements in both steps of the framework. On the one hand, we propose to encode node similarities based on the free energy distance, which interpolates between the shortest path and the commute time distances, thus, providing an additional degree of flexibility. On the other hand, we propose a matrix factorization method based on a loss function that generalizes that of the skip-gram model with negative sampling to arbitrary similarity matrices. Compared with factorizations based on the widely used$\ell _{2}$loss, the proposed method can better preserve node pairs associated with higher similarity scores. Moreover, it can be easily implemented using advanced automatic differentiation toolkits and computed efficiently by leveraging GPU resources. Node clustering, node classification, and link prediction experiments on real-world datasets demonstrate the effectiveness of incorporating free-energy-based similarities as well as the proposed matrix factorization compared with state-of-the-art alternatives.

We develop a novel graph-based trainable framework to maximize the weighted sum energy efficiency (WSEE) for power allocation in wireless communication networks. To address the non-convex nature of the problem, the proposed method consists of modular structures inspired by a classical iterative suboptimal approach and enhanced with learnable components. More precisely, we propose a deep unfolding of the successive concave approximation (SCA) method. In our unfolded SCA (USCA) framework, the originally preset parameters are now learnable via graph convolutional neural networks (GCNs) that directly exploit multi-user channel state information as the underlying graph adjacency matrix. We show the permutation equivariance of the proposed architecture, which is a desirable property for models applied to wireless network data. The USCA framework is trained through a stochastic gradient descent approach using a progressive training strategy. The unsupervised loss is carefully devised to feature the monotonic property of the objective under maximum power constraints. Comprehensive numerical results demonstrate its generalizability across different network topologies of varying size, density, and channel distribution. Thorough comparisons illustrate the improved performance and robustness of USCA over state-of-the-art benchmarks.

Efficient scheduling of transmissions is a key problem in wireless networks. The main challenge stems from the fact that optimal link scheduling involves solving a maximum weighted independent set (MWIS) problem, which is known to be NP-hard. In practical schedulers, centralized and distributed greedy heuristics are commonly used to approximately solve the MWIS problem. However, most of these greedy heuristics ignore important topological information of the wireless network. To overcome this limitation, we propose fast heuristics based on graph convolutional networks (GCNs) that can be implemented in centralized and distributed manners. Our centralized heuristic is based on tree search guided by a GCN and 1-step rollout. In our distributed MWIS solver, a GCN generates topology-aware node embeddings that are combined with per-link utilities before invoking a distributed greedy solver. Moreover, a novel reinforcement learning scheme is developed to train the GCN in a non-differentiable pipeline. Test results on medium-sized wireless networks show that our centralized heuristic can reach a near-optimal solution quickly, and our distributed heuristic based on a shallow GCN can reduce by nearly half the suboptimality gap of the distributed greedy solver with minimal increase in complexity. The proposed schedulers also exhibit good generalizability across graph and weight distributions.

Solving the optimal symbol detection problem in multiple-input multiple-output (MIMO) systems is known to be NP-hard. Hence, the objective of any detector of practical relevance is to get reasonably close to the optimal solution while keeping the computational complexity in check. In this work, we propose a MIMO detector based on an annealed version of Langevin (stochastic) dynamics. More precisely, we define a stochastic dynamical process whose stationary distribution coincides with the posterior distribution of the symbols given our observations. In essence, this allows us to approximate the maximum a posteriori estimator of the transmitted symbols by sampling from the proposed Langevin dynamic. Furthermore, we carefully craft this stochastic dynamic by gradually adding a sequence of noise with decreasing variance to the trajectories, which ensures that the estimated symbols belong to a pre-specified discrete constellation. Based on the proposed MIMO detector, we also design a robust version of the method by unfolding and parameterizing one term– the score of the likelihood– by a neural network. Through numerical experiments in both synthetic and real-world data, we show that our proposed detector yields state-of-the-art symbol error rate performance and the robust version becomes noise-variance agnostic.

Production systems use heuristics because they are faster or scale better than the corresponding optimal algorithms. Yet, practitioners are often unaware of how worse off a heuristic's solution may be with respect to the optimum in realistic scenarios. Leveraging two-stage games and convex optimization, we present a provable framework that unveils settings where a given heuristic underperforms.

Graph neural networks (GNNs) have achieved superior performance on node classification tasks in the last few years. Commonly, this is framed in a transductive semi-supervised learning setup wherein the entire graph – including the target nodes to be labeled – is available for training. Driven in part by scalability, recent works have focused on the inductive case where only the labeled portion of a graph is available for training. In this context, our current work considers a challenging inductive setting where a set of labeled graphs are available for training while the unlabeled target graph is completely separate, i.e., there are no connections between labeled and unlabeled nodes. Under the implicit assumption that the testing and training graphs come from similar distributions, our goal is to develop a labeling function that generalizes to unobserved connectivity structures. To that end, we employ a graph neural tangent kernel (GNTK) that corresponds to infinitely wide GNNs to find correspondences between nodes in different graphs based on both the topology and the node features. We augment the capabilities of the GNTK with residual connections and empirically illustrate its performance gains on standard benchmarks.

Power allocation is one of the fundamental problems in wireless networks and a wide variety of algorithms address this problem from different perspectives. A common element among these algorithms is that they rely on an estimation of the channel state, which may be inaccurate on account of hardware defects, noisy feedback systems, and environmental and adversarial disturbances. Therefore, it is essential that the output power allocation of these algorithms is stable with respect to input perturbations, to the extent that the variations in the output are bounded for bounded variations in the input. In this paper, we focus on UWMMSE – a modern algorithm leveraging graph neural networks –, and illustrate its stability to additive input perturbations of bounded energy through both theoretical analysis and empirical validation.

Particle filtering is used to compute nonlinear estimates of complex systems. It samples trajectories from a chosen distribution and computes the estimate as a weighted average of them. Easy-to-sample distributions often lead to degenerate samples where only one trajectory carries all the weight, negatively affecting the resulting performance of the estimate. While much research has been done on the design of appropriate sampling distributions that would lead to controlled degeneracy, in this paper our objective is to learn sampling distributions. Leveraging the framework of algorithm unrolling, we model the sampling distribution as a multivariate normal, and we use neural networks to learn both the mean and the covariance. We carry out unsupervised training of the model to minimize weight degeneracy, relying only on the observed measurements of the system. We show in simulations that the resulting particle filter yields good estimates in a wide range of scenarios.

We propose a data-driven approach for power allocation in the context of federated learning (FL) over interference-limited wireless networks. The power policy is designed to maximize the transmitted information during the FL process under communication constraints, with the ultimate objective of improving the accuracy and efficiency of the global FL model being trained. The proposed power allocation policy is parameterized using a graph convolutional network and the associated constrained optimization problem is solved through a primal-dual algorithm. Numerical experiments show that the proposed method outperforms three baseline methods in both transmission success rate and FL global performance.

We consider the problem of estimating the topology of multiple networks from nodal observations, where these networks are assumed to be drawn from the same (unknown) random graph model. We adopt a graphon as our random graph model, which is a nonparametric model from which graphs of potentially different sizes can be drawn. The versatility of graphons allows us to tackle the joint inference problem even for the cases where the graphs to be recovered contain different number of nodes and lack precise alignment across the graphs. Our solution is based on combining a maximum likelihood penalty with graphon estimation schemes and can be used to augment existing network inference methods. We validate our proposed approach by comparing its performance against competing methods in synthetic and real-world datasets.

Learning graphs from sets of nodal observations represents a prominent problem formally known as graph topology inference. However, current approaches are limited by typically focusing on inferring single networks, and they assume that observations from all nodes are available. First, many contemporary setups involve multiple related networks, and second, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by these facts, we introduce a joint graph topology inference method that models the influence of the hidden variables. Under the assumptions that the observed signals are stationary on the sought graphs and the graphs are closely related, the joint estimation of multiple networks allows us to exploit such relationships to improve the quality of the learned graphs. Moreover, we confront the challenging problem of modeling the influence of the hidden nodes to minimize their detrimental effect. To obtain an amenable approach, we take advantage of the particular structure of the setup at hand and leverage the similarity between the different graphs, which affects both the observed and the hidden nodes. To test the proposed method, numerical simulations over synthetic and real-world graphs are provided.

We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on simplicial complexes. Then, we refine this idea to construct wavelets that respect the Hodge-Helmholtz decomposition. For these Hodgelets, familiar notions of curl-free and divergence-free flows from vector calculus are preserved. We characterize the representational quality of our Hodgelets for edge flows in terms of frame bounds and demonstrate the use of these spectral wavelets for sparse representation of edge flows on real and synthetic data.

Distributed scheduling algorithms for throughput or utility maximization in dense wireless multi-hop networks can have overwhelmingly high overhead, causing increased congestion, energy consumption, radio footprint, and security vulnerability. For wireless networks with dense connectivity, we propose a distributed scheme for link sparsification with graph convolutional networks (GCNs), which can reduce the scheduling overhead while keeping most of the network capacity. In a nutshell, a trainable GCN module generates node embeddings as topology-aware and reusable parameters for a local decision mechanism, based on which a link can withdraw itself from the scheduling contention if it is not likely to win. In medium-sized wireless networks, our proposed sparse scheduler beats classical threshold-based sparsification policies by retaining almost 70% of the total capacity achieved by a distributed greedy max-weight scheduler with 0.4% of the point-to-point message complexity and 2.6% of the average number of interfering neighbors per link.

In wireless multi-hop networks, delay is an important metric for many applications. However, the max-weight scheduling algorithms in the literature typically focus on instantaneous optimality, in which the schedule is selected by solving a maximum weighted independent set (MWIS) problem on the interference graph at each time slot. These myopic policies perform poorly in delay-oriented scheduling, in which the dependency between the current backlogs of the network and the schedule of the previous time slot needs to be considered. To address this issue, we propose a delay-oriented distributed scheduler based on graph convolutional networks (GCNs). In a nutshell, a trainable GCN module generates node embeddings that capture the network topology as well as multi-step lookahead backlogs, before calling a distributed greedy MWIS solver. In small- to medium-sized wireless networks with heterogeneous transmit power, where a few central links have many interfering neighbors, our proposed distributed scheduler can outperform the myopic schedulers based on greedy and instantaneously optimal MWIS solvers, with good generalizability across graph models and minimal increase in communication complexity.

We propose a flexible framework for defining the 1-Laplacian of a hypergraph that incorporates edge-dependent vertex weights. These weights are able to reflect varying importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity than homogeneous hypergraphs. We then utilize the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian to cluster the vertices. From a theoretical standpoint based on an adequately defined normalized Cheeger cut, this procedure is expected to achieve higher clustering accuracy than that based on the traditional Laplacian. Indeed, we confirm that this is the case using real-world datasets to demonstrate the effectiveness of the proposed spectral clustering approach. Moreover, we show that for a special case within our framework, the corresponding hypergraph 1-Laplacian is equivalent to the 1-Laplacian of a related graph, whose eigenvectors can be computed more efficiently, facilitating the adoption on larger datasets.

Graph search is one of the most successful algorithmic trends in near neighbor search. Several of the most popular and empirically successful algorithms are, at their core, a greedy walk along a pruned near neighbor graph. However, graph traversal applications often suffer from poor memory access patterns, and near neighbor search is no exception to this rule. Our measurements show that popular search indices such as the hierarchical navigable small-world graph (HNSW) can have poor cache miss performance. To address this issue, we formulate the graph traversal problem as a cache hit maximization task and propose multiple graph reordering as a solution. Graph reordering is a memory layout optimization that groups commonly-accessed nodes together in memory. We mathematically formalize the connection between the graph layout and the cache complexity of search. We present exhaustive experiments applying several reordering algorithms to a leading graph-based near neighbor method based on the HNSW index. We find that reordering improves the query time by up to 40%, we present analysis and improvements for existing graph layout methods, and we demonstrate that the time needed to reorder the graph is negligible compared to the time required to construct the index.

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. Graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime.

We study the problem of optimal power allocation in a single-hop ad hoc wireless network. In solving this problem, we propose a hybrid neural architecture inspired by the algorithmic unfolding of the iterative weighted minimum mean squared error (WMMSE) method, that we denote as unfolded WMMSE (UWMMSE). The learnable weights within UWMMSE are parameterized using graph neural networks (GNNs), where the time-varying underlying graphs are given by the fading interference coefficients in the wireless network. These GNNs are trained through a gradient descent approach based on multiple instances of the power allocation problem. Once trained, UWMMSE achieves performance comparable to that of WMMSE while significantly reducing the computational complexity. This phenomenon is illustrated through numerical experiments along with the robustness and generalization to wireless networks of different densities and sizes.

We consider the problem of sequential graph topology change-point detection from graph signals. We assume that signals on the nodes of the graph are regularized by the underlying graph structure via a graph filtering model, which we then leverage to distill the graph topology change-point detection problem to a subspace detection problem. We demonstrate how prior information on the spectral signature of the post-change graph can be incorporated to implicitly denoise the observed sequential data, thus leading to a natural CUSUM-based algorithm for change-point detection. Numerical experiments illustrate the performance of our proposed approach, particularly underscoring the benefits of (potentially noisy) prior information.

We study the problem of adaptive contention window (CW) design for random-access wireless networks. More precisely, our goal is to design an intelligent node that can dynamically adapt its minimum CW (MCW) parameter to maximize a network-level utility knowing neither the MCWs of other nodes nor how these change over time. To achieve this goal, we adopt a reinforcement learning (RL) framework where we circumvent the lack of system knowledge with local channel observations and we reward actions that lead to high utilities. To efficiently learn these preferred actions, we follow a deep Q-learning approach, where the Q-value function is parametrized using a multi-layer perceptron. In particular, we implement a rainbow agent, which incorporates several empirical improvements over the basic deep Q-network. Numerical experiments based on the NS3 simulator reveal that the proposed RL agent performs close to optimal and markedly improves upon existing learning and non-learning based alternatives.

We consider the problem of inferring the unobserved edges of a graph from data supported on its nodes. In line with existing approaches, we propose a convex program for recovering a graph Laplacian that is approximately diagonalizable by a set of eigenvectors obtained from the second-order moment of the observed data. Unlike existing work, we incorporate prior knowledge about the distribution from where the underlying graph was drawn. In particular, we consider the case where the graph was drawn from a graphon model, and we supplement our convex optimization problem with a provably-valid regularizer on the spectrum of the graph to be recovered. We present the cases where the graphon model is assumed to be known and the more practical setting where the relevant features of the model are inferred from auxiliary network observations. Numerical experiments on synthetic and real-world data illustrate the advantage of leveraging the proposed graphon prior, even when the prior is imperfect.

A fundamental problem in the design of wireless networks is to efficiently schedule transmission in a distributed manner. The main challenge stems from the fact that optimal link scheduling involves solving a maximum weighted independent set (MWIS) problem, which is NP-hard. For practical link scheduling schemes, distributed greedy approaches are commonly used to approximate the solution of the MWIS problem. However, these greedy schemes mostly ignore important topological information of the wireless networks. To overcome this limitation, we propose a distributed MWIS solver based on graph convolutional networks (GCNs). In a nutshell, a trainable GCN module learns topology-aware node embeddings that are combined with the network weights before calling a greedy solver. In small- to middle-sized wireless networks with tens of links, even a shallow GCN-based MWIS scheduler can leverage the topological information of the graph to reduce in half the suboptimality gap of the distributed greedy solver with good generalizability across graphs and minimal increase in complexity.

We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture: namely, permutation equivariance, orientation equivariance, and simplicial awareness. The first two properties respectively account for the fact that the node indexing and the simplex orientations in a simplicial complex are arbitrary. The last property encodes the desirable feature that the output of the neural network depends on the entire simplicial complex and not on a subset of its dimensions. Based on these properties, we propose a simple convolutional architecture, rooted in tools from algebraic topology, for the problem of trajectory prediction, and show that it obeys all three of these properties when an odd, nonlinear activation function is used. We then demonstrate the effectiveness of this architecture in extrapolating trajectories on synthetic and real datasets, with particular emphasis on the gains in generalizability to unseen trajectories.

In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common higher-order network abstractions that can incorporate polyadic relationships. We provide brief introductions to simplicial complexes and hypergraphs, with a special emphasis on the concepts needed for the processing of signals supported on these structures. Specifically, we discuss Fourier analysis, signal denoising, signal interpolation, node embeddings, and nonlinear processing through neural networks, using these two higher-order network models. In the context of simplicial complexes, we specifically focus on signal processing using the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing. For hypergraphs, we present both matrix and tensor representations, and discuss the trade-offs in adopting one or the other. We also highlight limitations and potential research avenues, both to inform practitioners and to motivate the contribution of new researchers to the area.

We study the problem of optimal power allocation in a single-hop ad hoc wireless network. In solving this problem, we depart from classical purely model-based approaches and propose a hybrid method that retains key modeling elements in conjunction with data-driven components. More precisely, we put forth a neural network architecture inspired by the algorithmic unfolding of the iterative weighted minimum mean squared error (WMMSE) method, that we denote by unfolded WMMSE (UWMMSE). The learnable weights within UWMMSE are parameterized using graph neural networks (GNNs), where the time-varying underlying graphs are given by the fading interference coefficients in the wireless network. These GNNs are trained through a gradient descent approach based on multiple instances of the power allocation problem. We show that the proposed architecture is permutation equivariant, thus facilitating generalizability across network topologies. Comprehensive numerical experiments illustrate the performance attained by UWMMSE along with its robustness to hyper-parameter selection and generalizability to unseen scenarios such as different network densities and network sizes.

We study the problem of missing data imputation for graph signals from signed one-bit quantized observations. More precisely, we consider that the true graph data is drawn from a distribution of signals that are smooth or bandlimited on a known graph. However, instead of observing these signals, we observe a signed version of them and only at a subset of the nodes on the graph. Our goal is to estimate the true underlying graph signals from our observations. To achieve this, we propose a generative adversarial network (GAN) where the key is to incorporate graph-aware losses in the associated minimax optimization problem. We illustrate the benefits of the proposed method via numerical experiments on hand-written digits from the MNIST dataset.

We study the blind centrality ranking problem, where our goal is to infer the eigenvector centrality ranking of nodes solely from nodal observations, i.e., without information about the topology of the network. We formalize these nodal observations as graph signals and model them as the outputs of a network process on the underlying (unobserved) network. A simple spectral algorithm is proposed to estimate the leading eigenvector of the associated adjacency matrix, thus serving as a proxy for the centrality ranking. A finite rate performance analysis of the algorithm is provided, where we find a lower bound on the number of graph signals needed to correctly rank (with high probability) two nodes of interest. We then specialize our general analysis for the particular case of dense Erdös-Rényi graphs, where existing graph-theoretical results can be leveraged. Finally, we illustrate the proposed algorithm via numerical experiments on synthetic and real-world networks, with special emphasis on how the network features influence the performance.

In this paper, we consider the problem of projecting networks onto metric spaces. Networks are structures that encode relationships between pairs of elements or nodes. However, these relationships can be independent of each other, and need not be defined for every pair of nodes. This is in contrast to a metric space, which requires that a distance between every pair of elements in the space be defined. To understand how to project networks onto metric spaces, we take an axiomatic approach: we first state two axioms for projective maps from the set of all networks to the set of finite metric spaces, then show that only one projection satisfies these requirements. The developed technique is shown to be an effective method for finding approximate solutions to combinatorial optimization problems. Finally, we illustrate the use of metric trees for efficient search in projected networks.

We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this fashion, the filtered graph signals can be interpreted as defined on a time-varying network. We model each of the underlying network realizations as generated by an independent draw from a latent stochastic blockmodel (SBM). To infer the partition of the latent SBM, we propose a simple spectral algorithm for which we provide a theoretical analysis and establish consistency guarantees for the recovery. We illustrate our results using numerical experiments on synthetic and real data, highlighting the efficacy of our approach.

We study the problem of jointly estimating several network processes that are driven by the same input, recasting it as one of blind identification of a bank of graph filters. More precisely, we consider the observation of several graph signals - i.e., signals defined on the nodes of a graph - and we model each of these signals as the output of a different network process (represented by a graph filter) defined on a common known graph and driven by a common unknown input. Our goal is to recover the specifications of every network process by only observing the outputs. Since every process shares the same input, the estimation problems are coupled, and a joint inference method is proposed. We study two different scenarios, one where the orders of the filters are known, and one where they are not. For the former case we propose a least-squares approach and provide conditions for recovery. For the latter case, we put forth a sparse recovery algorithm with theoretical guarantees. Finally, we illustrate the methods here proposed via numerical experiments.

We present a graph-based semi-supervised learning (SSL) method for learning edge flows defined on a graph. Specifically, given flow measurements on a subset of edges, we want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flow conservation. These constraints render our approach different from classical graph-based SSL for vertex labels, which posits that tightly connected nodes share similar labels and leverages the graph structure accordingly to extrapolate from a few vertex labels to the unlabeled vertices. We derive bounds for our method's reconstruction error and demonstrate its strong performance on synthetic and real-world flow networks from transportation, physical infrastructure, and the Web. Furthermore, we provide two active learning algorithms for selecting informative edges on which to measure flow, which has applications for optimal sensor deployment. The first strategy selects edges to minimize the reconstruction error bound and works well on flows that are approximately divergence-free. The second approach clusters the graph and selects bottleneck edges that cross cluster-boundaries, which works well on flows with global trends.

This paper generalizes the classical joint problem of signal demixing and blind deconvolution to the realm of graphs. We investigate a setup where a single observation formed by the sum of multiple graph signals is available. The main assumption is that each individual signal is generated by an originally sparse input diffused through the graph via the application of a graph filter. In this context, we address the related problems of: 1) separating the individual graph signals, 2) identifying the unknown input supports, and 3) estimating the coefficients of the diffusing graph filters. We first consider the case where each signal - prior to mixing - is diffused in a different graph. We then particularize the results for the more challenging case where all the signals are diffused in the same graph. The corresponding demixing and blind graph-signal deconvolution problems are formulated, convex relaxations are presented, and recovery conditions are discussed. Numerical experiments in both the single and multiple graph cases show the capabilities of demixing in synthetic and biology-inspired graphs.

We address the problem of inferring an undirected graph from nodal observations, which are modeled as non-stationary graph signals generated by local diffusion dynamics on the unknown network. We propose a two-step approach where we first estimate the unknown diffusion (graph) filter, from which we recover the eigenvectors of the so-called graph-shift operator (a matrix representation of the graph). We then estimate the eigenvalues by imposing desirable properties on the graph to be recovered. To carry out the initial system identification step, we assume that second-order statistics of the inputs are available. While such quadratic filter identification problem boils down to a non-convex fourth order polynomial minimization, we propose a semidefinite relaxation with provable performance guarantees. Finally, numerical tests illustrate the use of the proposed algorithm to unveil urban mobility patterns.

This paper considers the problem of inferring the topology of a graph from noisy outputs of an unknown graph filter excited by low-rank signals. Limited by this low-rank structure, we focus on solving the community detection problem, whose aim is to partition the node set of the unknown graph into subsets with high edge densities. We propose to detect the communities by applying spectral clustering on the low-rank output covariance matrix. To analyze the performance, we show that the low-rank covariance yields a sketch of the eigenvectors of the unknown graph. Importantly, we provide theoretical bounds on the error introduced by this sketching procedure based on spectral features of the graph filter involved. Finally, our theoretical findings are validated via numerical experiments.

This paper investigates the problems of signal reconstruction and blind deconvolution for graph signals that have been generated by an originally sparse input diffused through the network via the application of a graph filter operator. Assuming that the support of the sparse input signal is unknown, and that the diffused signal is observed only at a subset of nodes, we address the related problems of: 1) identifying the input and 2) interpolating the values of the diffused signal at the non-sampled nodes. We first consider the more tractable case where the coefficients of the diffusing graph filter are known and then address the problem of joint input and filter identification. The corresponding blind identification problems are formulated, novel convex relaxations are discussed, and modifications to incorporate a priori information on the sparse inputs are provided.

Advancing a holistic theory of networks and network processes requires the extension of existing results in the processing of time-varying signals to signals supported on graphs. This paper focuses on the definition of stationarity and power spectral density for random graph signals, generalizes the concepts of autoregressive and moving average random processes to the graph domain, and investigates their parametric spectral estimation. Theoretical and algorithmic results are complemented with numerical tests on synthetic and real-world graphs.

We address the problem of identifying a graph structure from the observation of signals defined on its nodes. Fundamentally, the unknown graph encodes direct relationships between signal elements, which we aim to recover from observable indirect relationships generated by a diffusion process on the graph. We put forth a novel network topology inference approach whereby we: i) identify the eigenvectors of a matrix representation of the graph from realizations of the diffused signal; and ii) rely on these (possibly imperfect) spectral templates to estimate the eigenvalues by imposing desirable properties on the graph to be recovered. Robust algorithms with quantifiable performance are developed for the pragmatic settings where the eigenvectors are estimated with errors, or, when the eigenbasis is only partially known. Numerical tests showcase the effectiveness of the proposed algorithm in recovering amino-acid networks.

We address the problem of inferring a graph from nodal observations, which are modeled as non-stationary graph signals generated by local diffusion dynamics that depend on the structure of the sought network. Using the so-called graph-shift operator (GSO) as a matrix representation of the graph, we first identify the eigenvectors of the shift matrix from realizations of the diffused signals, and then we rely on these spectral templates to estimate the eigenvalues by imposing desirable properties on the graph to be recovered. Different from the stationary setting where the GSO and the covariance matrix of the observed signals are simultaneously diagonalizable, here they are not. Hence, estimating the eigenvectors requires first estimating the unknown diffusion (graph) filter - a polynomial in the GSO which does preserve the sought eigenbasis. To carry out this initial system identification step, we leverage different sources of information on the input signal driving the diffusion process on the graph. Numerical tests showcase the effectiveness of the proposed algorithms in recovering social and structural brain graphs.

We present a novel method to hierarchically cluster networked data allowing nodes to simultaneously belong to multiple clusters. Given a network, our method outputs a cut metric on the underlying node set, which can be related to data coverings at different resolutions. The cut metric is obtained by averaging a set of ultrametrics, which are themselves the output of (non-overlapping) hierarchically clustering noisy versions of the original network of interest. The resulting algorithm is illustrated in synthetic networks and is used to classify handwritten digits from the MNIST database.

This paper presents diffusion filtering as a method to smooth signals defined on the nodes of a graph or network. Diffusion filtering considers the given signals as initial temperature distributions in the nodes and diffuses heat through the edges of the graph. The filtered signal is determined by the accumulated temperatures over time at each node. We show multiple other interpretations of diffusion filtering and describe how it can be generalized to encompass a wide class of networks making it suitable for real-world applications. We prove that diffused signals are stable to perturbations in the underlying network. Further, we demonstrate how diffusion filtering can be applied to improve the performance of recommendation systems by considering the problem of predicting ratings from a signal processing perspective.

A novel scheme for sampling graph signals is proposed. Space-shift sampling can be understood as a hybrid scheme that combines selection sampling -- observing the signal values on a subset of nodes - and aggregation sampling - observing the signal values at a single node after successive aggregation of local data. Under the assumption of bandlimitedness, we state conditions and propose strategies for signal recovery in different settings. Being a more general procedure, space-shift sampling achieves smaller reconstruction errors than current schemes, as we illustrate through the reconstruction of the industrial activity in a graph of the U.S. economy.

Network processes are often represented as signals defined on the vertices of a graph. To untangle the latent structure of such signals, one can view them as outputs of linear graph filters modeling underlying network dynamics. This paper deals with the problem of joint identification of a graph filter and its input signal, thus broadening the scope of classical blind deconvolution of temporal and spatial signals to the less-structured graph domain. Given a graph signal y modeled as the output of a graph filter, the goal is to recover the vector of filter coefficients h, and the input signal x which is assumed to be sparse. While y is a bilinear function of x and h, the filtered graph signal is also a linear combination of the entries of the "lifted" rank-one, row-sparse matrix xhT. The blind graph filter identification problem can be thus tackled via rank and sparsity minimization subject to linear constraints, an approach amenable to convex relaxation. An algorithm for jointly processing multiple output signals corresponding to different sparse inputs is also developed. Numerical tests with synthetic and real-world networks illustrate the merits of the proposed algorithm, as well as the benefits of leveraging multiple signals to aid the blind identification task.

We introduce node-variant graph filters, which allow the simultaneous implementation of multiple (regular) graph filters at different nodes, and study their design to implement arbitrary linear transformations between graph signals. Node-variant graph filters can be implemented distributedly, making them suitable for networked settings. We determine spectral conditions under which a specific linear transformation can be implemented perfectly and, for the cases where perfect implementation is infeasible, the design of optimal approximations for different error metrics is analyzed. We demonstrate the practical relevance of the developed framework by studying the application of node-variant graph filters for analog network coding.

This paper introduces a formal definition of continuity and generalizes an existing notion of stability for node centrality measures in weighted graphs. It is shown that the frequently used measures of degree, closeness and eigenvector centrality are stable and continuous whereas betweenness centrality is neither. Numerical experiments in synthetic and real-world networks show that both stability and continuity are desirable in practice since they imply different levels of robustness in the presence of noisy data. In particular, a stable alternative of betweenness centrality is shown to exhibit resilience against noise while preserving its notion of centrality.

This paper presents a formal definition of stability for node centrality measures in networks and shows that the well-known betweenness centrality is not stable with respect to that metric. An alternative definition that preserves the same centrality notion while satisfying this stability criterion is then introduced. The practical implications of stability are explored by studying the behavior of the traditional as well as the alternative stable betweenness centrality in both, synthetic random networks, and the network of interactions between sectors of the United States economy.

This paper introduces hierarchical quasi-clustering methods, a generalization of hierarchical clustering for asymmetric networks where the output structure preserves the asymmetry of the input data. We show that this output structure is equivalent to a finite quasi-ultrametric space and study admissibility with respect to two desirable properties. We prove that a modified version of single linkage is the only admissible quasi-clustering method. Moreover, we show stability of the proposed method and we establish invariance properties fulfilled by it. Algorithms are further developed and the value of quasi-clustering analysis is illustrated with a study of internal migration within United States.

We present an axiomatic construction of hierarchical clustering in asymmetric networks where the dissimilarity from node a to node b is not necessarily equal to the dissimilarity from node b to node a. The theory is built on the axioms of value and transformation which encode desirable properties common to any clustering method. Two hierarchical clustering methods that abide to these axioms are derived: reciprocal and nonreciprocal clustering. We further show that any clustering method that satisfies the axioms of value and transformation lies between reciprocal and nonreciprocal clustering in a well defined sense. We apply this theory to the formation of circles of trust in social networks.

We present an authorship attribution method based on relational data between function words. These are content independent words that help define grammatical relationships. As relational structures we use normalized word adjacency networks. We interpret these networks as Markov chains and compare them using entropy measures. We illustrate the accuracy of the method developed through a series of numerical experiments including comparisons with frequency based methods. We show that accuracy increases when combining relational and frequency based data, indicating that both sources of information encode different aspects of authorial styles.