Previous research shows that for structured populations located on a graph, one of the most important attributes that determines whether a cooperative community is robust is the topology of the graph. However, even in a graph that is highly robust with respect to cooperation, "weak points" may still exist which will allow defection to spread quickly in the community. Previous work shows that the transitivity and the average degree are related to the robustness of cooperation in the entire graph. In addition to considering the cooperation level across the entire graph, whether an individual in the graph will allow the spread of defection is an important research question in its own right. In this work, we are trying to identify both the "weak" individuals and the "robust" ones. We measure the centrality in the graph together with the degree, the local clustering coefficient, the betweenness, the closeness, the degree eigenvector, and a few newly designed centrality measures such as "clustering eigenvector centrality". The results show that for graphs that have a fixed number of vertices and edges, there are both robust individuals and weak individuals and that the higher the transitivity of the graph, the more robust the individuals are in the graph. However, although some of the graph centrality measures may indicate whether a vertex is robust or not, the prediction is still quite unstable.