This work introduces a new centrality metric called DomiRank, which quantifies the dominance of nodes in their neighborhoods. DomiRank integrates both local and global topological information through a tunable parameter, making it applicable to large networks. The metric is defined as the stationary solution of a dynamical process involving competition among nodes, where the level of competition is controlled by the parameter $\sigma$. High values of DomiRank highlight nodes with fragile neighborhoods, making them critical to network integrity. The authors present an analytical formula and an efficient parallelizable algorithm for computing DomiRank, demonstrating its superior performance in generating targeted attacks that effectively compromise network structure and functionality. DomiRank-based attacks are shown to cause more enduring damage, hindering the network's ability to recover. The metric is validated on both synthetic and real-world networks, highlighting its broad applicability in various domains, including improving spam detection, establishing vaccination schemes, and assessing vulnerabilities in transportation networks.This work introduces a new centrality metric called DomiRank, which quantifies the dominance of nodes in their neighborhoods. DomiRank integrates both local and global topological information through a tunable parameter, making it applicable to large networks. The metric is defined as the stationary solution of a dynamical process involving competition among nodes, where the level of competition is controlled by the parameter $\sigma$. High values of DomiRank highlight nodes with fragile neighborhoods, making them critical to network integrity. The authors present an analytical formula and an efficient parallelizable algorithm for computing DomiRank, demonstrating its superior performance in generating targeted attacks that effectively compromise network structure and functionality. DomiRank-based attacks are shown to cause more enduring damage, hindering the network's ability to recover. The metric is validated on both synthetic and real-world networks, highlighting its broad applicability in various domains, including improving spam detection, establishing vaccination schemes, and assessing vulnerabilities in transportation networks.