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We investigate online network topology identification from smooth nodal observations acquired in a streaming fashion. Different from nonadaptive batch solutions, our distinctive goal is to track the (possibly) dynamic adjacency matrix with affordable memory and computational costs by processing signal snapshots online. To this end, we leverage and truncate dualbased proximal gradient (DPG) iterations to solve a composite smoothnessregularized, timevarying inverse problem. Numerical tests with synthetic and real electrocorticography data showcase the effectiveness of the novel lightweight iterations when it comes to tracking slowlyvarying network connectivity. We also show that the online DPG algorithm converges faster than a primalbased baseline of comparable complexity. Aligned with reproducible research practices, we share the code developed to produce all figures included in this paper.more » « lessFree, publiclyaccessible full text available June 4, 2024

Free, publiclyaccessible full text available June 4, 2024

Free, publiclyaccessible full text available June 1, 2024

Machine learning frameworks such as graph neural networks typically rely on a given, fixed graph to exploit relational inductive biases and thus effectively learn from network data. However, when said graphs are (partially) unobserved, noisy, or dynamic, the problem of inferring graph structure from data becomes relevant. In this paper, we postulate a graph convolutional relationship between the observed and latent graphs, and formulate the graph structure learning task as a network inverse (deconvolution) problem. In lieu of eigendecompositionbased spectral methods or iterative optimization solutions, we unroll and truncate proximal gradient iterations to arrive at a parameterized neural network architecture that we call a Graph Deconvolution Network (GDN). GDNs can learn a distribution of graphs in a supervised fashion, perform link prediction or edgeweight regression tasks by adapting the loss function, and they are inherently inductive as well as node permutation equivariant. We corroborate GDNâ€™s superior graph learning performance and its generalization to larger graphs using synthetic data in supervised settings. Moreover, we demonstrate the robustness and representation power of GDNs on real world neuroimaging and social network datasets.more » « lessFree, publiclyaccessible full text available June 1, 2024

Oh, Alice ; Agarwal, Alekh ; Belgrave, Danielle ; Cho, Kyunghyun (Ed.)Graph neural networks (GNN) are an effective framework that exploit interrelationships within graphstructured data for learning. Principal component analysis (PCA) involves the projection of data on the eigenspace of the covariance matrix and draws similarities with the graph convolutional filters in GNNs. Motivated by this observation, we study a GNN architecture, called coVariance neural network (VNN), that operates on sample covariance matrices as graphs. We theoretically establish the stability of VNNs to perturbations in the covariance matrix, thus, implying an advantage over standard PCAbased data analysis approaches that are prone to instability due to principal components associated with close eigenvalues. Our experiments on realworld datasets validate our theoretical results and show that VNN performance is indeed more stable than PCAbased statistical approaches. Moreover, our experiments on multiresolution datasets also demonstrate that VNNs are amenable to transferability of performance over covariance matrices of different dimensions; a feature that is infeasible for PCAbased approaches.more » « less

Advances in graph signal processing for network neuroscience offer a unique pathway to integrate brain structure and function, with the goal of revealing some of the brain's organizing principles at the system level. In this direction, we develop a supervised graph representation learning framework to model the relationship between brain structural connectivity (SC) and functional connectivity (FC) via a graph encoderdecoder system. Specifically, we propose a Siamese network architecture equipped with graph convolutional encoders to learn graph (i.e., subject)level embeddings that preserve applicationdependent similarity measures between brain networks. This way, we effectively increase the number of training samples and bring in the flexibility to incorporate additional prior information via the prescribed target graphlevel distance. While information on the brain structurefunction coupling is implicitly distilled via reconstruction of brain FC from SC, our model also manages to learn representations that preserve the similarity between input graphs. The superior discriminative power of the learnt representations is demonstrated in downstream tasks including subject classification and visualization. All in all, this work advocates the prospect of leveraging learnt graphlevel, similaritypreserving embeddings for brain network analysis, by bringing to bear standard tools of metric data analysis.more » « less

We propose a deep learning solution to the inverse problem of localizing sources of network diffusion. Invoking graph signal processing (GSP) fundamentals, the problem boils down to blind estimation of a diffusion filter and its sparse input signal encoding the source locations. While the observations are bilinear functions of the unknowns, a mild requirement on invertibility of the graph filter enables a convex reformulation that we solve via the alternatingdirection method of multipliers (ADMM). We unroll and truncate the novel ADMM iterations, to arrive at a parameterized neural network architecture for Source Localization on Graphs (SLoGNet), that we train in an endtoend fashion using labeled data. This way we leverage inductive biases of a GSP modelbased solution in a datadriven trainable parametric architecture, which is interpretable, parameter efficient, and offers controllable complexity during inference. Experiments with simulated data corroborate that SLoGNet exhibits performance in par with the iterative ADMM baseline, while attaining significant (posttraining) speedups.more » « less

The popular Random Dot Product Graph (RDPG) generative model postulates that each node has an associated (latent) vector, and the probability of existence of an edge between two nodes is their innerproduct (with variants to consider directed and weighted graphs). In any case, the latent vectors may be estimated through a spectral decomposition of the adjacency matrix, the socalled Adjacency Spectral Embedding (ASE). Assume we are monitoring a stream of graphs and the objective is to track the latent vectors. Examples include recommender systems or monitoring of a wireless network. It is clear that performing the ASE of each graph separately may result in a prohibitive computation load. Furthermore, the invariance to rotations of the inner product complicates comparing the latent vectors at different timesteps. By considering the minimization problem underlying ASE, we develop an iterative algorithm that updates the latent vectors' estimation as new graphs from the stream arrive. Differently to other proposals, our method does not accumulate errors and thus does not requires periodically recomputing the spectral decomposition. Furthermore, the pragmatic setting where nodes leave or join the graph (e.g. a new product in the recommender system) can be accommodated as well. Our code is available at https://github.com/marfiori/efficientASEmore » « less

The Random Dot Product Graph (RDPG) is a popular generative graph model for relational data. RDPGs postulate there exist latent positions for each node, and specifies the edge formation probabilities via the inner product of the corresponding latent vectors. The embedding task of estimating these latent positions from observed graphs is usually posed as a nonconvex matrix factorization problem. The workhorse Adjacency Spectral Embedding offers an approximate solution obtained via the eigendecomposition of the adjacency matrix, which enjoys solid statistical guarantees but can be computationally intensive and is formally solving a surrogate problem. In this paper, we bring to bear recent nonconvex optimization advances and demonstrate their impact to RDPG inference. We develop firstorder gradient descent methods to better solve the original optimization problem, and to accommodate broader network embedding applications in an organic way. The effectiveness of the resulting graph representation learning framework is demonstrated on both synthetic and real data. We show the algorithms are scalable, robust to missing network data, and can track the latent positions over time when the graphs are acquired in a streaming fashion.more » « less