## 1 Introduction
### Technical Approach and Paper Outline  
We tackle the network topology inference problem of the previous section in two fundamental settings. We start in Section 2.1 with the batch (offline) case where observations \(\{\mathbf{y}_{t}\}_{t=1}^{T}\) are all available for joint processing as in [6; 10]. Here though we exploit the implications of stationarity from a different angle which leads to a formulation amenable to efficient solutions via proximal gradient (PG) algorithms; see e.g., [11; 12]. Specifically, as we show in Section 2.1.1 stationarity implies that the covariance matrix \(\mathbf{C}_{\mathbf{y}}\) of the observations commutes with \(\mathbf{S}\) under mild requirements; see also [7]. This motivates formulating the topology inference task as an inverse problem, whereby one searches for a sparse \(\mathbf{S}\) that is structurally admissible and approximately commutes with the observations' empirical covariance matrix \(\hat{\mathbf{C}}_{\mathbf{y}}\). In [6; 10], the algorithmic issues were not thoroughly studied and relied mostly on off-the-shelf convex solvers whose scalability could be challenged. Unlike [13] but similar to link prediction problems ([1], Ch. 7.2), [14], here we rely on a priori knowledge about the presence (or absence) of a few edges; conceivably leading to simpler algorithmic updates and better recovery performance. We may learn about edge status via limited questionnaires and experiments, or, we could perform edge screening prior to topology inference [15]; see also the discussion in Section 4. The batch PG algorithm developed in Section 2.1.3 to recover the network topology is computationally more efficient than existing methods for this (and related) problem(s). It also serves as a fundamental building block to construct online iterations, which constitutes the second main contribution of this paper.  
Focus in Section 2.2 shifts to online recovery of \(\mathbf{S}\) from _streaming_ signals \(\{\mathbf{y}_{1},\ldots,\mathbf{y}_{t},\mathbf{y}_{t+1},\ldots\}\), each of them adhering to the generative model in (1). For streaming data, the empirical covariance estimate \(\hat{\mathbf{C}}_{\mathbf{y},t}\) can be updated recursively, and in Section 2.2.1 we show that online PG iterations can be brought to bear to efficiently track the time-varying solution of the inverse problem with quantifiable (non-asymptotic) guarantees. Different from [13], the updates are simple and devoid of multiple inner loops to compute expensive projections. Moreover, we establish convergence to within a neighborhood of the optimal time-varying batch solution as well as dynamic regret bounds (Section 2.2.2) by adapting results in [16]. The algorithm and analyses of this paper are valid even for dynamic networks, i.e., if the GSO \(\mathbf{S}_{t}\) in (1) is (slowly) time-varying. Indeed, we examine how the variability and eigenvectors of the underlying graph as well as the diffusion filters' frequency response influence the size of the convergence radius (or misadjustment in the adaptive filtering parlance). Numerical tests in Section 3 corroborate the efficiency and effectiveness of the proposed algorithm in adapting to streaming information and tracking changes in the sought dynamic network. A brief general discussion about the results obtained, the assumptions made, and the limitations, as well as the impact this work can have on applications, is presented in Section 4. Concluding remarks are given in Section 5.