Choosing a leader on a complex network

In many real life applications a group of people interact through a communication network, mathematically modelled as a connected graph linking each element of the group. These participants may have diverse objectives and play very different roles depending on their knowledge and privileges. We focus on a particular scenario, in which a certain node is absolutely essential for completing the intended task. Moreover, if a technical failure results in disconnection of a participant to this leader node, this participant can no longer take part in the group's performance.