The economic development and social well‐being of modern societies are highly dependent on networked critical infrastructures to satisfy the necessary demand. In the case that the prescribed demand cannot be satisfied with each component of a network normally loaded, certain components need to be overloaded. To study the influence of overloading on the network reliability, this research proposes a new flow‐redistribution rule based on minimal cuts to determine the components to overload when necessary and the way to redistribute the network flow. The proposed rule can help to keep the prescribed demand for a network satisfied, while trying to minimize the adverse influence of overloading on the network reliability. Based on the proposed rule, we then propose a bi‐objective optimization model to identify a network's critical components that, among all the possible combinations of components with the same amount, influence the network reliability the most when incapacitated, while considering the possibility of overloading and its influence on the reliability.