This study introduces DomiRank centrality, a new metric that quantifies the dominance of nodes in their neighborhoods, integrating local and global network information through a tunable parameter. DomiRank identifies nodes that are highly dominant in their local environment, which are critical for network integrity and functionality. The metric is defined as the stationary solution of a dynamical process, and an analytical formula is derived for efficient computation. DomiRank outperforms other centrality metrics in generating targeted attacks that effectively compromise network structure and disrupt functionality, both in synthetic and real-world networks. It also causes more enduring damage, hindering network recovery and impairing resilience. DomiRank is computationally efficient and can be applied to massive sparse networks using parallelizable algorithms on GPUs. The parameter σ controls the competition mechanism, influencing the balance between local and global network properties. The study shows that DomiRank-based attacks are highly effective in dismantling networks, particularly in synthetic and real-world topologies, and cause more severe damage than other centrality-based attacks. The results demonstrate that DomiRank is a powerful tool for identifying critical nodes in complex systems, enhancing network resilience, and informing mitigation strategies. The method is applicable to various domains, including transportation, social networks, and biological systems, and can be used to design effective vaccination schemes and other interventions. The study highlights the importance of understanding network fragility and the role of dominance in network structure and function.This study introduces DomiRank centrality, a new metric that quantifies the dominance of nodes in their neighborhoods, integrating local and global network information through a tunable parameter. DomiRank identifies nodes that are highly dominant in their local environment, which are critical for network integrity and functionality. The metric is defined as the stationary solution of a dynamical process, and an analytical formula is derived for efficient computation. DomiRank outperforms other centrality metrics in generating targeted attacks that effectively compromise network structure and disrupt functionality, both in synthetic and real-world networks. It also causes more enduring damage, hindering network recovery and impairing resilience. DomiRank is computationally efficient and can be applied to massive sparse networks using parallelizable algorithms on GPUs. The parameter σ controls the competition mechanism, influencing the balance between local and global network properties. The study shows that DomiRank-based attacks are highly effective in dismantling networks, particularly in synthetic and real-world topologies, and cause more severe damage than other centrality-based attacks. The results demonstrate that DomiRank is a powerful tool for identifying critical nodes in complex systems, enhancing network resilience, and informing mitigation strategies. The method is applicable to various domains, including transportation, social networks, and biological systems, and can be used to design effective vaccination schemes and other interventions. The study highlights the importance of understanding network fragility and the role of dominance in network structure and function.