Properties of the n‐overlap vector and n‐overlap similarity theory

In the first part of this article the author defines the n‐overlap vector whose coordinates consist of the fraction of the objects (e.g., books, N‐grams, etc.) that belong to 1, 2, …, n sets (more generally: families) (e.g., libraries, databases, etc.). With the aid of the Lorenz concentration theory, a theory of n‐overlap similarity is conceived together with corresponding measures, such as the generalized Jaccard index (generalizing the well‐known Jaccard index in case n 5 2). Next, the distributional form of the n‐overlap vector is determined assuming certain distributions of the object's and of the set (family) sizes. In this section the decreasing power law and decreasing exponential distribution is explained for the n‐overlap vector. Both item (token) n‐overlap and source (type) n‐overlap are studied. The n‐overlap properties of objects indexed by a hierarchical system (e.g., books indexed by numbers from a UDC or Dewey system or by N‐grams) are presented in the final section. The author shows how the results given in the previous section can be applied as well as how the Lorenz order of the n‐overlap vector is respected by an increase or a decrease of the level of refinement in the hierarchical system (e.g., the value N in N‐grams).