Lorenz Curves Determine Partial Orders for Comparing Network Structures

Abstract

Networks can be studied from different points of view. In this paper, it is shown that the traditional Lorenz curve and some of its generalisations can be used for characterising inequality in network properties. Each type of Lorenz curve determines a partial order in a set of networks and a Gini-type index can be associated to each of these curves. The following types of Lorenz curves, each related to a different type of inequality in network properties and resulting in a different partial order, are briefly discussed: (a) Classical Lorenz curves and evenness measures; (b) intrinsic diversity profiles (or k-dominance curves) and associated measures of diversity; (c) generalised Lorenz curves as introduced by Shorrocks, that are not scale invariant but take absolute numbers into account; (d) weighted Lorenz curves for comparisons with an internal or external standard; and (e) Lorenz-type curve introduced to perform source per source comparisons of items. This paper claims that the Lorenz curves and the Gini index are universal tools for studying inequality, including inequality in network properties.