Networks are known to be prone to link failures. In this paper we set out to investigate how networks of varying connectivity patterns respond to different link failure schemes in terms of connectivity, clustering coefficient and shortest path lengths. We then propose a measure, which we call the vulnerability of a network, for evaluating the extent of the damage these failures can cause. Accepting the disconnections of node pairs as a damage indicator, vulnerability simply represents how quickly the failure of the critical links cause the network to undergo a specified damage extent. Analyzing the vulnerabilities under varying damage specifications shows that scale free networks are relatively more vulnerable for small failures, but more efficient; whereas Erdös-Rényi networks are the least vulnerable despite lacking any clustered structure.