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We develop online graph learning algorithms from streaming network data. Ourgoal is to track the (possibly) time-varying network topology, and effectmemory and computational savings by processing the data on-the-fly as they areacquired. The setup entails observations modeled as stationary graph signalsgenerated by local diffusion dynamics on the unknown network. Moreover, we mayhave a priori information on the presence or absence of a few edges as in thelink prediction problem. The stationarity assumption implies that theobservations' covariance matrix and the so-called graph shift operator (GSO --a matrix encoding the graph topology) commute under mild requirements. Thismotivates formulating the topology inference task as an inverse problem,whereby one searches for a sparse GSO that is structurally admissible andapproximately commutes with the observations' empirical covariance matrix. Forstreaming data said covariance can be updated recursively, and we show onlineproximal gradient iterations can be brought to bear to efficiently track thetime-varying solution of the inverse problem with quantifiable guarantees.Specifically, we derive conditions under which the GSO recovery cost isstrongly convex and use this property to prove that the online algorithmconverges to within a neighborhood of the optimal time-varying batch solution.Numerical tests illustrate the effectiveness of the proposed graph learningapproach in adapting to streaming information and tracking changes in thesought dynamic network.

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