Relational diversity.

In biology, the measurement of diversity traditionally focusses on reporting number of unambiguously distinguishable types, thus referring to qualitative (discontinuously varying) traits. Inclusion of frequencies or other weights has produced a large variety of diversity indices. Quantitative (continuously varying) traits do not readily fit into this perspective. In fact, in the context of quantitative traits, the concept of diversity is not always clearly distinguished from the (statistical) notion of dispersion. In many cases the ambiguity even extends to qualitative traits. This is at variance with the broad spectrum of diversity issues ranging, e.g., from ecological and genetic aspects of diversity to functional, structural, systematic, or evolutionary (including phylogenetic) aspects. In view of the urgent need for a more consistent perspective, it is called to attention that all of these aspects, whether of qualitative or quantitative nature, can be gathered under the common roof of binary relations (for qualitative traits two objects are related, for example, if they share the same trait state). A comprehensive concept of (relational) diversity can be developed in two steps: (1) determine the number of unrelated pairs of objects among all admissible pairs as a measure of implicit (relative) diversity, (2) invoke the concept of effective number to transform the implicit measure of diversity into an explicit (absolute) measure. The transformation operates by equating the observed implicit diversity to the implicit diversity obtained for the ideal model of an equivalence relation with classes of equal size. The number of these classes specifies the effective number as an explicit measure of diversity. The wealth of problems that can be treated from this unified perspective is briefly addressed by classifying and interpreting established diversity indices in the light of relational diversity. Desirable applications to the above-mentioned aspects are specified with the help of types of relations such as order, hierarchical, and tree relations. Corresponding biological issues including taxonomic community diversity, mating system, food web, sociological, cladistic and phylogenetic, or hypercycle diversity are suggested for future consideration.

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