Graph neural network-based tail-risk pricing for cyber insurance against cyber-physical attacks in power systems

Abstract Cyber insurance transfers the residual cyber risk that remains after technical controls are installed, yet conventional pricing models rarely condition losses on grid topology, operating state, communication dependencies, and physical consequences. We develop a graph neural network (GNN) framework for pricing false-data-injection, denial-of-service, relay-spoofing, and coordinated cyber-physical attacks in power systems. The study-specific contribution combines four elements: paired attack and no-attack consequence simulation; an attributed spatio-temporal graph model for the conditional claim distribution; explicit expected-loss, value-at-risk, conditional-value-at-risk, uncertainty, and capital loadings; and a safety-screened investment policy that links verified controls to premium reductions. The evaluation uses the public one-minute false-data-injection records and network data for the IEEE 30-bus and IEEE 118-bus systems, together with simulated communication and relay attacks. Monetary outcomes are reported in 2026 US dollars. The spatio-temporal GNN reduces the relative VaR95 and CVaR95 errors to 10% and 13%, respectively. Its tail-risk premium produces a 102% loss ratio against a 100% adequacy target. Coupling pricing with safe reinforcement learning lowers expected energy not served under coordinated attacks from 85 to 38 MWh per year. These results show how topology-conditioned cyber-physical losses can support auditable insurance prices and preventive investment decisions.