It is impossible for a system or network to keep completely safe with the possibility of many new threats occurring at any moment. To analyze and solve these kinds of problems, this paper presents a mathematical programming problem, which adopts a novel metric called Degree of Disconnectivity (DOD) to evaluate the damage level and survivability of a network. To evaluate and analyze the robustness of a network for network operators, this problem is modeled as a mathematical programming problem. Here, an attacker applies his limited attack power intelligently to the targeted network. The objective of the attacker is to compromise nodes, resulting in disconnections of O-D pairs, to ensure that the proposed degree of disconnectivity metric reachs a given level. A Lagrangean relaxation-based algorithm is adopted to solve the proposed problem.