Large Deviations and Entropy-Regularized Control on Critical Random Graphs

<div xmlns="http://www.tei-c.org/ns/1.0"> We study rare-event suppression in decentralized stochastic systems evolving over sparse random graphs near critical connectivity. Motivated by applications in safe control and decision-making on fragile networks, we analyze the tradeoff between control effort and rare-event mitigation in systems governed by local stochastic dynamics and feedback control. Using tools from large deviation theory, we derive path-space rate functions for interacting diffusions and jump processes on critical Erdős-Rényi graphs in the uncontrolled and entropy-regularized controlled setting. Our results reveal a universal scaling law: the minimum control energy required to achieve exponential suppression of catastrophic events grows proportionally to the graph diameter. We prove sharp lower bounds on energy-to-decay tradeoffs, characterize feedback-induced rate function transformations, and expose a fundamental tension between fragility and controllability in networked systems at the phase transition threshold. </div>