Multivariate Recovery Dependency in Networks with Time Delays: Effects on Resilience and Scaling Laws

Abstract The recovery process in real-world systems needs to consider the combined effects of internal and external failures and is rarely instantaneous, typically involving inherent time delays.Here, we investigate the impact of a time delay, τ, on system recovery in dependency networks, that can be homogeneous or heterogeneous, with both internal and external recovery mechanisms. Through theoretical analysis and numerical simulations, our findings reveal that the system exhibits two critical thresholds, pc1 and pc2, associated to complete recovery and irreversible failure, respectively. These thresholds divide the network recovery states into three regions: fully recoverable, potentially recoverable and unrecoverable, where the first region exhibits the optimal recovery time cost, which is reflected as the critical window in the rescue measures. Additionally, we observe that the network size N and time delay τ have a significant impact on pc1, but no effect on pc2, and exhibit scaling and linear relationships, respectively, with the value |pc1-pc2|. However, when the system topology changes, the stability of pc2 is disrupted, exhibiting characteristics of a first-order phase transition. These findings highlight the importance of minimizing delays to enhance network resilience, providing valuable insights for designing robust recovery strategies in critical infrastructures.