Efficiency and Hierarchy in Networks Connecting Heterogeneous Nodes with Heterogeneous Links

Abstract We address the question of efficiency in a connections model where both nodes and links are heterogeneous. First, efficiency is shown to require a hierarchical organization, coupling the values of the nodes and their visibility. Second, continuity of the technology is shown to guarantee the existence of an efficient network. Third, the architecture of efficient networks is identified. Efficiency is achieved by networks with a single non-trivial component, within which one of the nodes with the highest value is directly connected to the others; some pairs of which may be directly connected so that only the links to the central node generate externalities, while every link between peripheral nodes is only used by the two nodes that it connects and solely to see each other. This is proved by showing that an optimal matching of pairs of nodes and the links which form any given network, i.e. a matching that maximizes the value of the resulting network, yields a network with this structure.