We investigate properties for allocation rules on directed communication networks and the formation of such networks under these payoff properties. We study allocation rules satisfying two appealing properties, Component Efficiency (CE) and the Hierarchical Payoff Property (HPP). We show that such allocation rules exist if and only if we restrict ourselves to a class of weakly hierarchical networks. Strengthening the hierarchical payoff property provides a similar result regarding the class of (strongly) hierarchical networks. Such hierarchical networks possess an explicit top-down structure. Subsequently, we discuss several possibilities to model the formation of such hierarchical networks.
Slikker, M., Gilles, R. P., Norde, H. W., & Tijs, S. H. (2005). Directed networks, allocation properties and hierarchy formation. Mathematical Social Sciences, 49(1), 55-80. https://doi.org/10.1016/j.mathsocsci.2004.04.005