The problem of efficiently identifying critical nodes that substantially degrade network performance if they do not function is crucial and essential in analyzing a large complex network such as social networks on the Web, and it is still challenging. In this paper, we tackle this problem under a realistic situation where each link is probabilistically disconnected reflecting that an information path between two persons in a social network is not always open to pass on a message, rather than assuming that every information path is always open and passes on any information from one to the other. To solve this problem, we focus on the articulation point and utilize the bridge detection technique in graph theory to efficiently identify critical nodes in case the node reachability is taken as the performance measure. This corresponds to the total number of people who can receive information issued by every single person in a social network. Using two real-world social networks, we empirically show that the proposed method has a good scalability with respect to the network size and the nodes our method identified possesses unique properties and they are difficult to be identified by using conventional centrality measures.