Integrating Fragmented Risk Knowledge: Sheaf Theory for Risk Analysts.

As sociotechnical systems grow more interconnected, data-rich, and intricately distributed across organizational and disciplinary boundaries and over levels of organizational and technology hierarchies, there is an increasing need for risk assessment and risk management modeling tools that describe how local information, constraints, and decisions interact to shape global outcomes. "Systems-of-systems" (SoS) provide a starting point, but additional methods are needed to understand how risk assessments and management decisions based on partial, local knowledge can (or cannot) be integrated into globally coherent strategies. This paper explores applications of a powerful branch of mathematics, sheaf theory, as a mathematically principled framework for reasoning about local-to-global relationships, structural consistency, and context-dependent interpretation and integration of risk-related information and messages in complex systems. We seek to present an accessible, risk analyst-oriented introduction to core technical concepts and to illustrate the practical value of these mathematical techniques for risk analysis through examples drawn from risk psychology and decision-making under uncertainty; fusion of sensor data and other partial information; causal modeling and quantitative risk assessment of complex engineering and biological systems; risk communication and Social Amplification of Risk (SARF); collective and distributed planning and policy coordination; and ongoing monitoring and evaluation of threats and intervention outcomes. We show how sheaf-theoretic methods can detect irreconcilable policy conflicts, identify gaps in information fusion, simulate belief propagation under constraints, and support modular, multi-level modeling. Applications span risk communication, environmental monitoring, emergency planning, and regulatory governance. We conclude with a discussion of how sheaf theory might be integrated into mainstream risk science.