Partial orders for zero-sum arrays with applications to network theory

•We introduce several partial orders in the set of zero-sum arrays.•This leads to theories of hierarchy and domination.•We introduce a framework to compare power structures.•This theory is applied to directed, acyclic networks.

In this contribution we study partial orders in the set of zero-sum arrays. Concretely, these partial orders relate to local and global hierarchy and dominance theories. The exact relation between hierarchy and dominance curves is explained. Based on this investigation we design a new approach for measuring dominance or stated otherwise, power structures, in networks. A new type of Lorenz curve to measure dominance or power is proposed, and used to illustrate intrinsic characteristics of networks. The new curves, referred to as D-curves are partly concave and partly convex. As such they do not satisfy Dalton's transfer principle. Most importantly, this article introduces a framework to compare different power structures as a whole.It is shown that D-curves have several properties making them suitable to measure dominance. If dominance and being a subordinate are reversed, the dominance structure in a network is also reversed.