Discovering network resilience formulas with symbolized reinforcement learning

Many complex networks display remarkable resilience under external perturbations, internal failures and environmental changes, yet they can swiftly deteriorate into dysfunction upon the removal of a few keystone nodes. Discovering formulas that measure network resilience offers the potential to prevent catastrophic collapses—from species extinctions to financial crises—with profound implications for real-world systems. Existing resilience metrics are typically designed to characterize topological properties, whereas in many real systems resilience is defined by the dynamics imposed on the network; consequently, metrics that explicitly incorporate dynamical state information are expected to be better suited for identifying keystone nodes in such settings. Here, we report an automatic method for resilience metric discovery, which learns from how machine learning model solves a complicated network dismantling problem and symbolizes its network attack strategies into mathematical formulas. This proposed self-inductive approach discovers a resilience formula that accounts for both topology and dynamics, highlighting how the correlation between node degree and state shapes overall network resilience. Additionally, our approach discovers formulas that refine existing well-established resilience metrics with over 37.5% improvement in accuracy, significantly advancing human understanding of complex networks with machine learning. Complex networks can fail when a small number of critical components are lost. Here, the authors use symbolized reinforcement learning to derive interpretable formulas that identify these components and quantify network resilience.