Diversity (mathematics)

In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper, who call diversities "a form of multi-way metric". The concept finds application in nonlinear analysis.

Given a set $$X$$, let $$ \wp_\mbox{fin}(X)$$ be the set of finite subsets of $$X$$. A diversity is a pair $$(X,\delta)$$ consisting of a set $$X$$ and a function $$\delta \colon \wp_\mbox{fin}(X) \to \mathbb{R}$$ satisfying

(D1) $$\delta(A)\geq 0$$, with $$\delta(A)=0$$ if and only if $$\left|A\right|\leq 1$$

and

(D2) if $$ B\neq\emptyset$$ then $$\delta(A\cup C)\leq\delta(A\cup B) + \delta(B \cup C)$$.

Bryant and Tupper observe that these axioms imply monotonicity; that is, if $$A\subseteq B$$, then $$\delta(A)\leq\delta(B)$$. They state that the term "diversity" comes from the appearance of a special case of their definition in work on phylogenetic and ecological diversities. They give the following examples:

Diameter diversity
Let $$(X,d)$$ be a metric space. Setting $$\delta(A)=\max_{a,b\in A} d(a,b)=\operatorname{diam}(A)$$ for all $$A\in\wp_\mbox{fin}(X)$$ defines a diversity.

$$L_1$$ diversity
For all finite $$A\subseteq\mathbb{R}^n$$ if we define $$\delta(A)=\sum_i\max_{a,b}\left\{\left| a_i-b_i\right|\colon a,b\in A\right\}$$ then $$(\mathbb{R}^n,\delta)$$ is a diversity.

Phylogenetic diversity
If T is a phylogenetic tree with taxon set X. For each finite $$A\subseteq X$$, define $$\delta(A)$$ as the length of the smallest subtree of T connecting taxa in A. Then $$(X, \delta)$$ is a (phylogenetic) diversity.

Steiner diversity
Let $$(X, d)$$ be a metric space. For each finite $$A\subseteq X$$, let $$\delta(A)$$ denote the minimum length of a Steiner tree within X connecting elements in A. Then $$(X,\delta)$$ is a diversity.

Truncated diversity
Let $$(X,\delta)$$ be a diversity. For all $$A\in\wp_\mbox{fin}(X)$$ define $$\delta^{(k)}(A) = \max\left\{\delta(B)\colon |B|\leq k, B\subseteq A\right\}$$. Then if $$k\geq 2$$, $$(X,\delta^{(k)})$$ is a diversity.

Clique diversity
If $$(X,E)$$ is a graph, and $$\delta(A)$$ is defined for any finite A as the largest clique of A, then $$(X,\delta)$$ is a diversity.