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1. Online Network Topology Inference with Partial Connectivity Information

   https://doi.org/10.1109/CAMSAP45676.2019.9022677  

   Shafipour, Rasoul; Mateos, Gonzalo (December 2019, 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP))

   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. 

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2. Online Topology Inference from Streaming Stationary Graph Signals with Partial Connectivity Information

   https://doi.org/10.3390/a13090228  

   Shafipour, Rasoul; Mateos, Gonzalo (September 2020, Algorithms)

   null (Ed.)

   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. 

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3. Online proximal gradient for learning graphs from streaming signals

   https://doi.org/10.23919/Eusipco47968.2020.9287550  

   Shafipour, Rasoul; Mateos, Gonzalo (January 2021, 2020 28th European Signal Processing Conference (EUSIPCO))

   null (Ed.)

   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. 

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4. Online Graph Learning under Smoothness Priors

   https://doi.org/10.23919/EUSIPCO54536.2021.9616123  

   Saboksayr, Seyed Saman; Mateos, Gonzalo; Cetin, Mujdat (August 2021, 2021 29th European Signal Processing Conference (EUSIPCO))

   The growing success of graph signal processing (GSP) approaches relies heavily on prior identification of a graph over which network data admit certain regularity. However, adaptation to increasingly dynamic environments as well as demands for real-time processing of streaming data pose major challenges to this end. In this context, we develop novel algorithms for online network topology inference given streaming observations assumed to be smooth on the sought graph. Unlike existing batch algorithms, our goal is to track the (possibly) time-varying network topology while maintaining the memory and computational costs in check by processing graph signals sequentially-in-time. To recover the graph in an online fashion, we leverage proximal gradient (PG) methods to solve a judicious smoothness-regularized, time-varying optimization problem. Under mild technical conditions, we establish that the online graph learning algorithm converges to within a neighborhood of (i.e., it tracks) the optimal time-varying batch solution. Computer simulations using both synthetic and real financial market data illustrate the effectiveness of the proposed algorithm in adapting to streaming signals to track slowly-varying network connectivity. 

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5. DIRECTED NETWORK TOPOLOGY INFERENCE VIA GRAPH FILTER IDENTIFICATION

   https://doi.org/10.1109/DSW.2018.8439888  

   Shafipour, Rasoul; Segarra, Santiago; Marques, Antonio G.; Mateos, Gonzalo (June 2018, 2018 IEEE Data Science Workshop (DSW))

   We address the problem of inferring a directed network from nodal observations of graph signals generated by linear diffusion dynamics on the sought graph. Observations are modeled as the outputs of a linear graph filter (i.e., a polynomial on a local diffusion graph-shift operator encoding the unknown graph topology), excited with an ensemble of independent graph signals with arbitrarily-correlated nodal components. In this context, we first rely on observations of the output signals along with prior statistical information on the inputs to identify the diffusion filter. Such problem entails solving a system of quadratic matrix equations, which we recast as a smooth quadratic minimization subject to Stiefel manifold constraints. Subsequent identification of the network topology given the graph filter estimate boils down to finding a sparse and structurally admissible shift that commutes with the given filter, thus forcing the latter to be a polynomial in the sought graph-shift operator. Preliminary numerical tests corroborating the effectiveness of the proposed algorithms in recovering synthetic and real-world digraphs are provided. 

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