Consensus State Gram Matrix Estimation for Stochastic Switching Networks from Spectral Distribution Moments

Reaching distributed average consensus quickly and accurately over a network through iterative dynamics represents an important task in numerous distributed applications. Suitably designed filters applied to the state values can significantly improve the convergence rate. For constant networks, these filters can be viewed in terms of graph signal processing as polynomials in a single matrix, the consensus iteration matrix, with filter response evaluated at its eigenvalues. For random, time-varying networks, filter design becomes more complicated, involving eigendecompositions of sums and products of random, time-varying iteration matrices. This paper focuses on deriving an estimate for the Gram matrix of error in the state vectors over a filtering window for large-scale, stationary, switching random networks. The result depends on the moments of the empirical spectral distribution, which can be estimated through Monte-Carlo simulation. This work then defines a quadratic objective function to minimize the expected consensus estimate error norm. Simulation results provide support for the approximation.Comment: 52nd Asilomar Conference on Signals, Systems, and Computers (Asilomar 2017

Optimal Graph Filter Design for Large-Scale Random Networks

<p>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.</p