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.

L₁ 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.