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A Power network is a type of contact network where its members can tell other people what to do.
A political party is the most obvious example. It is typically much larger than a social network but is not one, since people can only call on each other for a small number of reasons, and perhaps at specific times. Also not every member of the network has the same role or right to help.
A corporation may also be a power network if it relies on many power relations to maintain itself - for instance, the right to draw oil from the ground or dump waste without being held responsible for it.
When a power network achieves some stable grasp of political power and daily working control of some important infrastructure, it becomes a power structure. In a democracy the visible power networks represented by the parties compete regularly in an election - but much of the power structure, typically the bureaucracy, does not change, and is typically ignored as a power network in itself.