Development of hierarchy theory for digraphs using concentration theory based on a new type of Lorenz curve

In digraphs one has a hierarchy based on the unidirectional order between the vertices of the graph. We present a method of measuring degrees of hierarchy as expressed by the inequality that exists between the vertices' hierarchical numbers. In order to do so, we need to extend the classical Lorenz theory of concentration (curves and measures) for a set of numbers x(1),..., x(N) to the case that Sigma(i=1)(N) x(i)=0. This is then applied to the set of hierarchical numbers of the vertices of the graph. A graph has a more concentrated hierarchy than another one if the Lorenz curve of the first one is above the Lorenz curve of the second one, hereby expressing that the inequality in domination in the first case is larger than in the second case, and that the inequality in subordination in the first case is larger than in the second case. We also determine maximal and minimal Lorenz curves in this setting and characterize the graphs that yield these curves. Based on this theory, we also determine good measures of hierarchical concentration in graphs. Applications can be given in the study of organigrams in companies and administrations and in citation analysis. (C) 2002 Elsevier Science Ltd. All rights reserved.