Resilience centrality in complex networks.

Resilience describes a system's ability to adjust its activity to retain the basic functionality when errors or failures occur in components (nodes) of the network. Due to the complexity of a system's structure, different components in the system exhibit diversity in the ability to affect the resilience of the system, bringing us a great challenge to protect the system from collapse. A fundamental problem is therefore to propose a physically insightful centrality index, with which to quantify the resilience contribution of a node in any systems effectively. However, existing centrality indexes are not suitable for the problem because they only consider the network structure of the system and ignore the impact of underlying dynamic characteristics. To break the limits, we derive a new centrality index: resilience centrality from the 1D dynamic equation of systems, with which we can quantify the ability of nodes to affect the resilience of the system accurately. Resilience centrality unveils the long-sought relations between the ability of nodes in a system's resilience and network structure of the system: the capacity is mainly determined by the degree and weighted nearest-neighbor degree of the node, in which weighted nearest-neighbor degree plays a prominent role. Further, we demonstrate that weighted nearest-neighbor degree has a positive impact on resilience centrality, while the effect of the degree depends on a specific parameter, average weighted degree β_{eff}, in the 1D dynamic equation. To test the performance of our approach, we construct four real networks from data, which corresponds to two complex systems with entirely different dynamic characteristics. The simulation results demonstrate the effectiveness of our resilience centrality, providing us theoretical insights into the protection of complex systems from collapse.