Graph signal processing analyzes signals supported on the nodes of a network with respect to a shift operator matrix that conforms to the graph structure. For shift-invariant graph filters, which are polynomial functions of the shift matrix, the filter response is defined by the value of the filter polynomial at the shift matrix eigenvalues. Thus, information regarding the spectral decomposition of the shift matrix plays an important role in filter design. However, under stochastic conditions leading to uncertain network structure, the eigenvalues of the shift matrix become random, complicating the filter design task. In such case, empirical distribution functions built from the random matrix eigenvalues may exhibit deterministic limiting behavior that can be exploited for problems on large-scale random networks. Acceleration filters for distributed average consensus dynamics on random networks provide the application covered in this thesis work. The thesis discusses methods from random matrix theory appropriate for analyzing adjacency matrix spectral asymptotics for both directed and undirected random networks, introducing relevant theorems. Network distribution properties that allow computational simplification of these methods are developed, and the methods are applied to important classes of random network distributions. Subsequently, the thesis presents the main contributions, which consist of optimization problems for consensus acceleration filters based on the obtained asymptotic spectral density information. The presented methods cover several cases for the random network distribution, including both undirected and directed networks as well as both constant and switching random networks. These methods also cover two related optimization objectives, asymptotic convergence rate and graph total variation.

We study a scalable approach to information fusion for large sensor networks. The algorithm, field inversion by consensus and compressed sensing (FICCS), is a distributed method for detection, localization, and estimation of a propagating field generated by an unknown number of point sources. The approach combines results in the areas of distributed average consensus and compressed sensing to form low dimensional linear projections of all sensor readings throughout the network, allowing each node to reconstruct a global estimate of the field. Compressed sensing is applied to continuous source localization by quantizing the potential locations of sources, transforming the model of sensor observations to a finite discretized linear model. We study the effects of structured modeling errors induced by spatial quantization and the robustness of ℓ1 penalty methods for field inversion. We develop a perturbations method to analyze the effects of spatial quantization error in compressed sensing and provide a model-robust version of noise-aware basis pursuit with an upperbound on the sparse reconstruction error. Numerical simulations illustrate system design considerations by measuring the performance of decentralized field reconstruction, detection performance of point phenomena, comparing trade-offs of quantization parameters, and studying various sparse estimators. The method is extended to time-varying systems using a recursive sparse estimator that incorporates priors into ℓ1 penalized least squares. This thesis presents the advantages of inter-sensor measurement mixing as a means of efficiently spreading information throughout a network, while identifying sparse estimation as an enabling technology for scalable distributed field reconstruction systems.