A small set of critical hyper-motifs governs heterogeneous flow-weighted network resilience

Flow-weighted networks are widespread in real-world systems, capturing the essence of flow interactions among various entities. Examples are food webs, social networks, transportation systems, and financial transactions. These networks are vulnerable to degradation when subjected to disturbances, often triggering cascading failures that severely impact their functionality. Despite their importance and recent advancements, the underlying mechanisms driving network degradation—from functional to dysfunctional states due to structural changes—remain poorly understood. In this study, we present a resilience analysis framework for flow-weighted networks. Our approach begins with constructing a hypergraph that encodes cascading failures through hyperedges. We then apply percolation theory to examine phase transitions and identify stable hyper-motifs throughout the degradation process. Our numerical simulations demonstrate that this framework discovers the Black Swan nodes in flow-weighted networks and provides a comprehensive resilience assessment. Our resilience analysis framework offers theoretical support for enhancing network resilience, suppressing rumor spread, preventing economic collapses, reducing traffic congestion, and improving ecological stability—ultimately fostering a more resilient and sustainable society.