Orthogonal Modal Representation in Long-Term Risk Quantification for Dynamic Multi-Agent Systems

Quantifying long-term risk in large-scale multi-agent systems is critical for ensuring safe operation. However, the high dimensionality of these systems and the rarity of risk events can make the required computations prohibitively expensive. To overcome this challenge, we introduce a graph-based representation and efficient risk quantification techniques tailored for stochastic multi-agent systems. A key technical innovation is a systematic approach to decompose the estimation problem of system-wide safety probabilities into smaller, lower-dimensional sub-systems with sub-safe sets. This decomposition leverages the graph Fourier basis of the agent interaction network, providing a natural and scalable representation. The safety probabilities for these sub-systems are derived as solutions to a set of low-dimensional partial differential equations (PDEs). The proposed decomposition enables existing risk quantification approaches but does so without an exponential increase in computational complexity with respect to the number of agents.