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This paper introduces B-GRASP, a Bayesian statistical framework for inferring uncertain edge weights in graph networks from noisy observations of node states in stochastic dynamical systems. The method connects graph-based models to partial differential equations (PDEs) and stochastic PDEs, incorporating diffusion and reaction dynamics with stochastic forcing to jointly estimate both graph structure and uncertainty. The framework is validated on heat conduction problems, reaction-diffusion systems, and real-world COVID-19 transmission data across U.S. states, demonstrating that full uncertainty quantification reveals information about network connectivity that point estimates alone cannot capture.
Why it matters
The framework has practical applications for understanding network dynamics in situations where connection strengths are uncertain, such as disease transmission networks, infrastructure systems, and other complex systems where accurate uncertainty quantification is critical for decision-making and risk assessment.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We present B-GRASP (Bayesian GRAph inference with SPDE priors), a Bayesian framework for inferring uncertain edge weights in stochastic dynamical systems on graphs from noisy observations of nodal states. The unknown edge weights parameterize the graph differential operator and therefore directly govern the evolution of the graph process. Motivated by connections between differential operators in PDEs and SPDEs and their graph counterparts, we construct stochastic graph models incorporating diffusion, reaction dynamics, and stochastic forcing. The resulting hierarchical formulation jointly represents uncertainty in the graph structure and stochastic forcing. Latent graph variables determine positive edge weights and the corresponding graph Laplacian, while latent Brownian variables represent the stochastic forcing. Conditional on these variables, the graph dynamics define a deterministic forward map from which the likelihood and posterior distribution are constructed. We characterize the posterior using maximum a posteriori estimation and the No-U-Turn Sampler, enabling both point estimation and uncertainty quantification. We demonstrate the framework on a one-dimensional inverse heat-conduction problem, stationary and nonstationary graph reaction-diffusion systems with nonlinear dynamics, and state-level COVID-19 data in the United States. The numerical results show that posterior uncertainty provides information not captured by point estimates, particularly for weakly identifiable or highly conductive edges, and enables uncertainty in both graph connectivity and stochastic forcing to be quantified within a Bayesian framework.
Source: B-GRASP: A Bayesian Framework for Inferring Graph Weights from SPDE-Inspired Dynamics