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 ${\mathrm{\wp }}_{\text{fin}}(X)$ be the set of finite subsets of $X$. A diversity is a pair $(X,\delta )$ consisting of a set $X$ and a function $\delta :{\mathrm{\wp }}_{\text{fin}}(X)\to \mathbb{ℝ}$ satisfying (D1) $\delta (A)\ge 0$, with $\delta (A)=0$ if and only if $\left|A\right|\le 1$ and (D2) if $B\ne \mathrm{\varnothing }$ then $\delta (A\cup C)\le \delta (A\cup B)+\delta (B\cup C)$. Bryant and Tupper observe that these axioms imply monotonicity; that is, if $A\subseteq B$, then $\delta (A)\le \delta (B)$. 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 $\delta (A)=\underset{a,b\in A}{\max }d(a,b)=\mathrm{diam}(A)$ for all $A\in {\mathrm{\wp }}_{\text{fin}}(X)$ defines a diversity.

L₁ diversity

For all finite $A\subseteq {\mathbb{ℝ}}^n$ if we define $\delta (A)=\sum _i\underset{a,b}{\max }\{|a_i-b_i|:a,b\in A\}$ 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 $A\in {\mathrm{\wp }}_{\text{fin}}(X)$ define ${\delta }^{(k)}(A)=\max \{\delta (B):|B|\le k,B\subseteq A\}$. Then if $k\ge 2$, $(X,{\delta }^{(k)})$ 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.