Measuring Polarization and Convergence as Transitional Processes in the Absence of a Cardinal Ordering

Conceptually Polarization and Convergence, objects of study in a variety of fields, are dynamic processes relating to specific types of transition between departure and arrival state distributions. Indeed the axiomatic development of polarization indices has been couched in terms of the impact on the shape of a consequent "final" distribution of cardinally measurable changes in locations and spreads of components of an initial distribution. The resultant indices end up as "distance weighted" summary statistics of the anatomy of the "final" distribution. However Polarization and Convergence concepts often pertain to situations where measurement is not cardinal. For example in many applications in the social sciences the departure and arrival states, which may be quite different in nature, frequently have just an ordinal ranking (e.g. social class departure state – economic or educational outcome arrival state). Such states are defined over one or more groups of agents and the dynamic processes are usually concerned with realignments of said agents within and between groupings. Here it is argued that in such situations polarization/convergence issues are more conveniently analyzed in the context of the anatomy of transitions between states which do not of necessity depend upon a between or within group cardinal ordering. Accordingly indices are proposed which are based upon the structure of an underlying transition process rather than the structure of the final state distribution. The measures do not depend upon the existence of a cardinal ordering but can be augmented to incorporate cardinality if such a metric is available. They do not depend upon the "square-ness" of the transition matrix, that is to say they can deal with disappearing and emerging groups. 3 examples from Canadian Generational Education Data, the world size distribution of Gross National Product per capita and Chinese Class Structures illustrate their use.