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,^[1] who call diversities "a form of multi-way metric".^[2] The concept finds application in nonlinear analysis.^[3]

This article is about a generalization of a metric space. For the measure frequently used in ecology, see Diversity index. For other uses of diversity, see Diversity.

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

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

and

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

Bryant and Tupper observe that these axioms imply monotonicity; that is, if ${\displaystyle A\subseteq B}$, then ${\displaystyle \delta (A)\leq \delta (B)}$. 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 ${\displaystyle (X,d)}$ be a metric space. Setting ${\displaystyle \delta (A)=\max _{a,b\in A}d(a,b)=\operatorname {diam} (A)}$ for all ${\displaystyle A\in \wp _{\mbox{fin}}(X)}$ defines a diversity.

L1 diversity

For all finite ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ if we define ${\displaystyle \delta (A)=\sum _{i}\max _{a,b}\left\{\left|a_{i}-b_{i}\right|\colon a,b\in A\right\}}$ then ${\displaystyle (\mathbb {R} ^{n},\delta )}$ is a diversity.

Phylogenetic diversity

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

Steiner diversity

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

Truncated diversity

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

Clique diversity

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