## 1 Introduction
### Contributions in Context of Prior Related Work  
Early topology inference approaches in the statistics literature can be traced back to the problem of (undirected) graphical model selection ([1], Ch. 7), [4; 5]. Under Gaussianity assumptions, this line of work has well-documented connections with covariance selection [17] and sparse precision matrix estimation [18; 19; 20], as well as neighborhood-based sparse linear regression [21]. Recent GSP-based network inference frameworks postulate that the network exists as a latent underlying structure, and that observations are generated as a result of a network process defined in such a graph [6; 10; 22; 23; 24; 25].  
Different from [22; 24; 26; 27; 28; 29] that infer structure from signals assumed to be smooth over the sought undirected graph, here the measurements are assumed related to the graph via filtering (cf. (1) and the opening discussion in Section 2.1). Few works have recently built on this rationale to identify a symmetric GSO given its eigenvectors, either assuming that the input is white [6; 10]--equivalently implying \(\mathbf{y}\) is graph stationary [7; 8; 9]; or, colored [30; 31]. Unlike prior online algorithms developed based on the aforementioned graph spectral domain design [13; 14], here we estimate the (possibly) time-varying GSO directly (without tracking its eigenvectors) and derive quantifiable recoveryguarantees; see Remark 3. Recent algorithms for identifying topologies of time-varying graphs [32; 33] operate in batch mode, they are non-recursive and hence their computational complexity grows linearly with time. While we assume that the graph signals are stationary, the online graph learning scheme in [34] uses observations from a Laplacian-based, continuous-time graph process. Unlike [35] that relies on a single-pole graph filter [36], the filter structure underlying (1) can be arbitrary, but the focus here is on learning undirected graphs. Online PG methods were adopted for directed graph inference under dynamic structural equation models [37], but lacking a formal performance analysis. The interested reader is referred to [38] for a comprehensive survey about topology identification of dynamic graphs with directional links. The recovery guarantees in Section 2.2.2 are adapted from the results in [16], obtained therein in the context of online sparse subspace clustering. Convergence and dynamic regret analysis techniques have been recently developed to study solutions of time-varying convex optimization problems; see [39] for a timely survey of this body of work. The impact of these optimization advances to dynamic network topology identification is yet to fully materialize, and this paper offers the first exploration in this direction.  
In closing, we note that information processing from streams of network data has been considered in recent work that falls under the umbrella of _adaptive_ GSP [40; 41]. Most existing results pertain to online reconstruction of partially-observed streaming graph signals, which are assumed to be bandlimited with respect to a known graph. An exception is [42], which puts forth an online algorithm for joint inference of the dynamic network topology and processes from partial nodal observations.