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We leverage proximal gradient iterations to develop an online graph learning algorithm from streaming network data. Our goal is to track the (possibly) time-varying network topology, and effect memory and computational savings by processing the data on-the-fly as they are acquired. The setup entails observations modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Moreover, we may have a priori information on the presence or absence of a few edges as in the link prediction problem. The stationarity assumption implies that the observations' covariance matrix and the so-called graph shift operator (GSO - a matrix encoding the graph topology) commute under mild requirements. This motivates formulating the topology inference task as an inverse problem, whereby one searches for a (e.g., sparse) GSO that is structurally admissible and approximately commutes with the observations' empirical covariance matrix. For streaming data said covariance can be updated recursively, and we show online proximal gradient iterations can be brought to bear to efficiently track the time-varying solution of the inverse problem with quantifiable guarantees. Specifically, we derive conditions under which the GSO recovery cost is strongly convex and use this property to prove that the online algorithm converges to within a neighborhood of the optimal time-varying batch solution. Preliminary numerical tests illustrate the effectiveness of the proposed graph learning approach in adapting to streaming information and tracking changes in the sought dynamic network. 

We develop online graph learning algorithms from streaming network data. Our goal is to track the (possibly) time-varying network topology, and affect memory and computational savings by processing the data on-the-fly as they are acquired. The setup entails observations modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Moreover, we may have a priori information on the presence or absence of a few edges as in the link prediction problem. The stationarity assumption implies that the observations’ covariance matrix and the so-called graph shift operator (GSO—a matrix encoding the graph topology) commute under mild requirements. This motivates formulating the topology inference task as an inverse problem, whereby one searches for a sparse GSO that is structurally admissible and approximately commutes with the observations’ empirical covariance matrix. For streaming data, said covariance can be updated recursively, and we show online proximal gradient iterations can be brought to bear to efficiently track the time-varying solution of the inverse problem with quantifiable guarantees. Specifically, we derive conditions under which the GSO recovery cost is strongly convex and use this property to prove that the online algorithm converges to within a neighborhood of the optimal time-varying batch solution. Numerical tests illustrate the effectiveness of the proposed graph learning approach in adapting to streaming information and tracking changes in the sought dynamic network. 

In this paper, we propose a supervised graph representation learning method to model the relationship between brain functional connectivity (FC) and structural connectivity (SC) through a graph encoder-decoder system. The graph convolutional network (GCN) model is leveraged in the encoder to learn lower-dimensional node representations (i.e. node embeddings) integrating information from both node attributes and network topology. In doing so, the encoder manages to capture both direct and indirect interactions between brain regions in the node embeddings which later help reconstruct empirical FC networks. From node embeddings, graph representations are learnt to embed the entire graphs into a vector space. Our end-to-end model utilizes a multi-objective loss function to simultaneously learn node representations for FC network reconstruction and graph representations for subject classification. The experiment on a large population of non-drinkers and heavy drinkers shows that our model can provide a characterization of the population pattern in the SC-FC relationship, while also learning features that capture individual uniqueness for subject classification. The identified key brain subnetworks show significant between-group difference and support the promising prospect of GCN-based graph representation learning on brain networks to model human brain activity and function. 

We develop algorithms for online topology inference from streaming nodal observations and partial connectivity information; i.e., a priori knowledge on the presence or absence of a few edges may be available as in the link prediction problem. The observations are modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Said stationarity assumption implies the simultaneous diagonalization of the observations' covariance matrix and the so-called graph shift operator (GSO), here the adjacency matrix of the sought graph. When the GSO eigenvectors are perfectly obtained from the ensemble covariance, we examine the structure of the feasible set of adjacency matrices and its dependency on the prior connectivity information available. In practice one can only form an empirical estimate of the covariance matrix, so we develop an alternating algorithm to find a sparse GSO given its imperfectly estimated eigenvectors. Upon sensing new diffused observations in the streaming setting, we efficiently update eigenvectors and perform only one (or a few) online iteration(s) of the proposed algorithm until a new datum is observed. Numerical tests showcase the effectiveness of the novel batch and online algorithms in recovering real-world graphs. 

In this paper, the relationship between functional and structural brain networks is investigated by training a graph encoder-decoder system to learn the mapping from brain structural connectivity (SC) to functional connectivity (FC). Our work leverages a graph convolutional network (GCN) model in the encoder which integrates both nodal attributes and the network topology information to generate new graph representations in lower dimensions. Using brain SC graphs as inputs, the novel GCN-based encoder-decoder system manages to account for both direct and indirect interactions between brain regions to reconstruct the empirical FC networks. In doing so, the latent variables within the system (i.e., the learnt low-dimensional embeddings) capture important information regarding the relation between functional and structural networks. By decomposing the reconstructed functional networks in terms of the output of each graph convolution filter, we can extract those brain regions which contribute most to the generation of FC networks from their SC counterparts. Experiments on a large population of healthy subjects from the Human Connectome Project show our model can learn a generalizable and interpretable SC-FC relationship. Overall, results here support the promising prospect of using GCNs to discover more about the complex nature of human brain activity and function. 

We address the problem of learning a sparsifying graph Fourier transform (GFT) for compressible signals on directed graphs (digraphs). Blending the merits of Fourier and dictionary learning representations, the goal is to obtain an orthonormal basis that captures spread modes of signal variation with respect to the underlying network topology, and yields parsimonious representations of bandlimited signals. Accordingly, we learn a data-adapted dictionary by minimizing a spectral dispersion criterion over the achievable frequency range, along with a sparsity-promoting regularization term on the GFT coefficients of training signals. An iterative algorithm is developed which alternates between minimizing a smooth objective over the Stiefel manifold, and soft-thresholding the graph-spectral domain representations of the signals in the training set. A frequency analysis of temperature measurements recorded across the contiguous United States illustrates the merits of the novel GFT design. 

We propose a methodology to carry out vertex-frequency analyses of graph signals, with the goal of unveiling the signal’s frequency occupancy over a localized region in the network. To this end, we first introduce localized graph signals in the vertex domain, by defining windows that are localized around each node by construction. Recent directed graph Fourier transform (DGFT) advances facilitate the frequency analysis of said localized signals, to reveal the signal’s energy distribution in a way akin to a spectrogram in the vertex-frequency plane. We then learn a set of windows by applying gradient descent method to an optimization problem governed by penalty parameters in the spectral domain. We also argue about the tradeoff between the resolution in the vertex and frequency domains based on the said parameters. We evaluate the performance of the proposed windowed GFT approach through numerical experiments on synthetic and real-world graphs. 

We address the problem of online topology inference from streaming nodal observations of graph signals generated by linear diffusion dynamics on the sought graph. To that end, we leverage the stationarity of the signals and use the so-called graph-shift operator (GSO) as a matrix representation of the graph. Under this model, estimated covariance eigenvectors obtained from streaming independent graph signals diffused on the sought network are a valid estimator of the GSO's spectral templates. We develop an ADMM algorithm to find a sparse and structurally admissible GSO given the eigenvectors estimate. Then, we propose an online scheme that upon sensing new diffused observations, efficiently updates eigenvectors (thus makes more accurate on expectation) and performs only one or a few iterations of the mentioned ADMM until the new data is observed. Numerical tests illustrate the effectiveness of the proposed topology inference approach in recovering large scale graphs, adapting to streaming information, and accommodating changes in the sought network.