  Inspired by socio-political scenarios, like dictatorships, in which a
minority of people exercise control over a majority of weakly interconnected
individuals, we propose vulnerability and power measures defined on groups of
actors of networks. We establish an unexpected connection between network
vulnerability and graph regularizability. We use the Shapley value of coalition
games to introduce fresh notions of vulnerability and power at node level
defined in terms of the corresponding measures at group level. We investigate
the computational complexity of computing the defined measures, both at group
and node levels, and provide effective methods to quantify them. Finally we
test vulnerability and power on both artificial and real networks.

Social and Information Networks;Physics and Society