Diversity (mathematics)

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In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper,{{cite journal|last1=Bryant|first1=David|last2=Tupper|first2=Paul|journal=Advances in Mathematics|volume=231|pages=3172–3198|year=2012|title=Hyperconvexity and tight-span theory for diversities|issue=6 |doi=10.1016/j.aim.2012.08.008|doi-access=free|arxiv=1006.1095}}

who call diversities "a form of multi-way metric".{{cite journal|last1=Bryant|first1=David|last2=Tupper|first2=Paul|journal=Discrete Mathematics and Theoretical Computer Science|title=Diversities and the geometry of hypergraphs|volume=16|issue=2|year=2014|pages=1–20|arxiv=1312.5408}} The concept finds application in nonlinear analysis.{{cite journal|last1=Espínola|first1=Rafa|last2=Pia̧tek|first2=Bożena|journal=Nonlinear Analysis|volume=95|year=2014|pages=229–245|title=Diversities, hyperconvexity, and fixed points|doi=10.1016/j.na.2013.09.005|hdl=11441/43016 |s2cid=119167622 |hdl-access=free}}

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{{sub|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.

References

Category:Metric spaces