Note from lecture: The signal sent by the context is sent graphlab, which then performs an update. The user does not write the code receiving the signal.

Is it only possible for a part of the graph to converge if it is disconnected from the rest of the graph?

None of the code posted on any of the slides is parallel code. It is just regular C/C++ functions. However, GraphLab processes multiple vertices in parallel. GraphLab can call scatter or gather on multiple vertices in parallel. GraphLab can run the code in parallel in any way it wants so long as it doesn't violate the scope dependencies declared by the application.

If the update to a vertex caused a significant change, then the surrounding vertices probably need another iteration of page rank to make sure that they have converged. This is a dynamic version of page rank where you continue to update regions of the graph until they have converged.