When do supply risks add up? Dependence-aware aggregation of critical-mineral risks in alloys and multi-element systems
摘要
Critical-mineral supply chains are increasingly exposed to geographic concentration, geopolitical shocks, and structural interdependence. Yet supply-risk indicators are usually defined for individual elements, whereas industrial and policy decisions often concern multi-element systems that fail when any required input becomes unavailable. Additive aggregation is therefore a useful baseline, but its validity depends on the dependence structure linking constituent minerals. We introduce a dependence-aware framework for aggregating elemental supply-risk scores across alloys and other multi-element systems. Element-level scores are related to latent incident probabilities through a low-incident first-order approximation. Cross-element dependence is modeled with a Gaussian copula whose correlation structure is constructed from two observable supply-chain drivers: shared country exposure and co-production relationships. Cascading host–by-product linkages are propagated through an absorbing Markov-chain formulation. Using the European Commission’s Supply Risk indicator as a worked example, we show that additive aggregation is often reasonable but can overstate system-level risk when several elements share upstream vulnerabilities. Risk then grows in a clustered and often concave manner: adding an element matters most when it activates a new dependence cluster. The framework provides a common relative scale for comparing pure elements and multi-element systems and can support industrial screening, substitution analysis, and the identification of supply-chain vulnerabilities requiring deeper investigation.