Bayesian Graph Signal Estimation in Nonlinear GSP Models with Multiple Topologies

Dynamic systems with evolving graph topologies arise in applications such as power grids and sensor networks. Conventional graph signal estimation methods often assume fixed topologies, limiting their applicability in dynamic environments. In this paper, we address the problem of Bayesian graph signal estimation in nonlinear models with varying networks by leveraging graph filters from graph signal processing (GSP) theory. We use the criterion of averaged mean-squared error (AMSE) across topologies, and develop the GSP-minimum linear AMSE (GSP-MLAMSE) estimator, which minimizes the AMSE among graph-filter-based estimators. We demonstrate that the GSP-MLAMSE estimator extends the GSP linear minimum mean-squared-error (GSP-LMMSE) estimator [1] to the case of multiple topologies. In addition, we prove that it achieves the minimum linear AMSE estimator for orthogonal graph frequencies. We also develop its parametric version via the Chebyshev graph filter, which ensures numerical stability and scalability while maintaining robustness to topology variations. Simulations of power system state estimation demonstrate significant improvements in the AMSE and robustness across varying signal-to-noise ratios (SNRs) compared to existing methods.