Efficiency-fragility discontinuity: Dual thresholds and the deceptive safety zone in critical infrastructure networks

Modern critical infrastructure networks optimize for maximum efficiency under stationary conditions, yet increasingly face nonstationary physical environments. We demonstrate through large-scale Monte Carlo simulation (1.55 million cascade events across five network scales) that efficiency-driven optimization drives supply networks toward a complex, two-stage phase transition. By coupling a spatially correlated stress model with topological simulation on Barabási-Albert scale-free networks, we identify two distinct critical thresholds: a continuous onset near η_{onset}≈0.79 marking the emergence of significant cascades, and a catastrophic threshold η_{sys}≈0.97 marking the discontinuous onset of systemic "dragon king" events. Between these thresholds lies a "deceptive safety zone" where operators observe frequent but contained failures, potentially normalizing risk while remaining unaware of proximity to total collapse. Finite-size scaling across N∈{10^{3},2.5×10^{3},5×10^{3},10^{4},2.5×10^{4}} shows the catastrophic threshold is essentially size independent (varying by ≈0.004 across a 25× range in N) while the onset converges as N^{-1/2} to η_{onset}(∞)=0.786±0.002, characteristic of a continuous transition. Pilot simulations on Watts-Strogatz and Erdős-Rényi topologies demonstrate that the dual-threshold gap persists universally, with threshold positions scaling monotonically with degree heterogeneity. We validate the framework against five historical infrastructure failures, demonstrating that recoverable events align with the robust regime while catastrophic collapses cluster near η_{sys}. Model comparison (ΔAIC=7057) decisively favors a bimodal dragon king distribution over a unimodal alternative at high efficiency.