Diversity (mathematics)

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In mathematics, a diversity is a way to measure how spread out a finite group of points is, generalizing the idea of distance from a metric. It assigns a nonnegative number to every finite subset of a universe X.

Definition and rules
- A diversity is a pair (X, δ) where δ maps every finite subset A of X to a real number.
- D1: δ(A) is always nonnegative, and δ(A) = 0 exactly when A has at most one element.
- D2: For any finite sets A, B, C with B nonempty, δ(A ∪ C) ≤ δ(A ∪ B) + δ(B ∪ C). This is a triangle-like rule.
- These rules imply monotonicity: if A ⊆ B then δ(A) ≤ δ(B).

Why the name diversity
- The term comes from applications in phylogenetic and ecological diversity, where people study how different a group of species is.

Examples
- Diameter diversity: If d is a usual distance on X, define δ(A) as the maximum distance between any two points in A. This is the diameter of A.
- Coordinate spread: For finite A ⊆ R^n, define δ(A) as the sum over coordinates of the largest difference in that coordinate among points of A. This measures spread across all coordinates.
- Phylogenetic tree diversity: If T is a tree with leaves X, δ(A) is the total length of the smallest subtree of T that connects all elements of A.
- Steiner-tree diversity: In a metric space (X, d), δ(A) is the minimum length of a Steiner tree inside X that connects A.
- k-subset diversity: For k ≥ 2, define δ^(k)(A) as the maximum δ(B) over all subsets B of A with at most k elements. Then (X, δ^(k)) is a diversity.
- Clique diversity: If X is a graph and δ(A) is the size of the largest clique contained in A, then (X, δ) is a diversity.

In short, diversities extend the idea of distance to quantify how diverse or spread out any finite group of points can be, using many natural ways to measure that spread.