In mathematics, a diversity is a generalization of the concept of a 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
{\displaystyle X}
, let
℘
fin
(
X
)
{\displaystyle \wp _{\mbox{fin}}(X)}
be the set of finite subsets of
X
{\displaystyle X}
.
A diversity is a pair
(
X
,
δ
)
{\displaystyle (X,\delta )}
consisting of a set
X
{\displaystyle X}
and a function
δ
:
℘
fin
(
X
)
→
R
{\displaystyle \delta \colon \wp _{\mbox{fin}}(X)\to \mathbb {R} }
satisfying
(D1)
δ
(
A
)
≥
0
{\displaystyle \delta (A)\geq 0}
, with
δ
(
A
)
=
0
{\displaystyle \delta (A)=0}
if and only if
|
A
|
≤
1
{\displaystyle \left|A\right|\leq 1}
and
(D2) if
B
≠
∅
{\displaystyle B\neq \emptyset }
then
δ
(
A
∪
C
)
≤
δ
(
A
∪
B
)
+
δ
(
B
∪
C
)
{\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
A
⊆
B
{\displaystyle A\subseteq B}
, then
δ
(
A
)
≤
δ
(
B
)
{\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:
Contents
Diameter diversity
Let
(
X
,
d
)
{\displaystyle (X,d)}
be a metric space. Setting
δ
(
A
)
=
max
a
,
b
∈
A
d
(
a
,
b
)
=
diam
(
A
)
{\displaystyle \delta (A)=\max _{a,b\in A}d(a,b)=\operatorname {diam} (A)}
for all
A
∈
L1 diversity
For all finite
A
⊆
R
n
{\displaystyle A\subseteq \mathbb {R} ^{n}}
if we define
δ
(
A
)
=
∑
i
max
a
,
b
{
|
a
i
−
b
i
|
:
a
,
b
∈
A
}
{\displaystyle \delta (A)=\sum _{i}\max _{a,b}\left\{\left|a_{i}-b_{i}\right|\colon a,b\in A\right\}}
then
Phylogenetic diversity
If T is a phylogenetic tree with taxon set X. For each finite
A
⊆
X
{\displaystyle A\subseteq X}
, define
δ
(
A
)
{\displaystyle \delta (A)}
as the length of the smallest subtree of T connecting taxa in A. Then
(
X
,
δ
)
{\displaystyle (X,\delta )}
is a (phylogenetic) diversity.
Steiner diversity
Let
(
X
,
d
)
{\displaystyle (X,d)}
be a metric space. For each finite
A
⊆
X
{\displaystyle A\subseteq X}
, let
δ
(
A
)
{\displaystyle \delta (A)}
denote
the minimum length of a Steiner tree within X connecting elements in A. Then
(
X
,
δ
)
{\displaystyle (X,\delta )}
is a
diversity.
Truncated diversity
Let
(
X
,
δ
)
{\displaystyle (X,\delta )}
be a diversity. For all
A
∈
℘
fin
(
X
)
{\displaystyle A\in \wp _{\mbox{fin}}(X)}
define
δ
(
k
)
(
A
)
=
max
{
δ
(
B
)
:
|
B
|
≤
Clique diversity
If
(
X
,
E
)
{\displaystyle (X,E)}
is a graph, and
δ
(
A
)
{\displaystyle \delta (A)}
is defined for any finite A as the largest clique of A, then
(
X
,
δ
)
{\displaystyle (X,\delta )}
is a diversity.