Unveiling Uniform Shifted Power Law in Stochastic Human and Autonomous Driving Behavior

Accurately simulating rare but safety-critical driving behaviors is essential for the evaluation and certification of autonomous vehicles (AVs). However, current models often fail to reproduce realistic collision rates when calibrated on real-world data, largely due to inadequate representation of long-tailed behavioral distributions. Here, we uncover a simple yet unifying shifted power law that robustly characterizes the stochasticity of both human-driven vehicle (HV) and AV behaviors, especially in the long-tail regime. The model adopts a parsimonious analytical form with only one or two parameters, enabling efficient calibration even under data sparsity. Analyzing large-scale, micro-level trajectory data from global HV and AV datasets, the shifted power law achieves an average R2 of 0.97 and a nearly identical tail distribution, uniformly fits both frequent behaviors and rare safety-critical deviations, significantly outperforming existing Gaussian-based baselines. When integrated into an agent-based traffic simulator, it enables forward-rolling simulations that reproduce realistic crash patterns for both HVs and AVs, achieving rates consistent with real-world statistics and improving the fidelity of safety assessment without post hoc correction. This discovery offers a unified and data-efficient foundation for modeling high-risk behavior and improves the fidelity of simulation-based safety assessments for mixed AV/HV traffic. The shifted power law provides a promising path toward simulation-driven validation and global certification of AV technologies.