# Gromov–Hausdorff class: its completeness and cloud geometry

S. A. Bogaty, A. A. Tuzhilin

###### Abstract

The paper is devoted to the study of the Gromov–Hausdorff proper class,
consisting of all metric spaces considered up to isometry. In this class, a
generalized Gromov–Hausdorff pseudometric is introduced and the geometry of
the resulting space is investigated. The first main result is a proof of the
completeness of the space, i.e., that all fundamental sequences converge in
it. Then we partition the space into maximal proper subclasses consisting of
spaces at a finite distance from each other. We call such subclasses clouds. A
multiplicative similarity group operates on clouds, multiplying all the
distances of each metric space by some positive number. We present examples of
similarity mappings transferring some clouds into another ones. We also show
that if a cloud contains a space that remains at zero distance from itself
under action of all similarities, then such a cloud contracted to this space.
In the final part, we investigate subsets of the real line with respect to
their behavior under various similarities.

## Introduction

This work is devoted to the study of the geometry of the Gromov–Hausdorff
proper class [1], which consists of all non-empty metric spaces considered up
to isometry. This class is endowed with the famous Gromov–Hausdorff distance
[2, 3, 4], which turns the class into a generalized pseudometric space
(“generalize” means that infinite distances may occur, and the prefix
“pseudo-” that the distances between non-isometric spaces may vanish).
Traditionally, this distance is studied on the Gromov–Hausdorff space, in
which all metric spaces are compact [5]. For non-compact spaces, a modified
distance is usually introduced — the pointed Gromov–Hausdorff distance [6]. In
this case, one chooses some points in metric spaces and makes sure that when
the spaces are compared with each other to define the distance, the points
turn out to be “close” to each other. However, it is more traditional for non-
compact spaces to consider the corresponding convergence without pre-setting
the distance function [5].

The decision to “break the tradition” was implemented in [7], where the
unmodified Gromov–Hausdorff distance is studied without restriction on
compactness. In the present work, we continue this study. It was noted in [7]
that although a pseudometric can be defined on a proper class in the standard
way, nevertheless, it is not possible to introduce the topology directly.
Indeed, recall that in von Neumann–Bernays–Gödel set theory, all objects are
called classes that belong to one of two types: sets and proper classes. A
class is called a set if it is a member of some other class. If the class is
not a member of any other class, then it is called proper. Thus, it is not be
possible to determine a topology on a proper class, because otherwise this
class itself is an element of the topology and, therefore, must be a set.
Nevertheless, for classes in which the cardinality filtering consists of sets,
a “topology” can be introduced by assigning a certain topology to each element
of the filtering, and requiring a natural matching of these topologies. This
is how the concept of a topological class was introduced in [7]. Also in [7]
it was shown that the Gromov–Hausdorff class is filtered by sets and,
therefore, the Gromov–Hausdorff distance allows us to turn this class into a
topological one.

The presence of such a “topology” allows one to define a continuous mapping
from a topological space to a topological class111The presence of a
pseudometric makes it possible to determine the continuity of a mapping from a
topological space in the standard way, using spherical neighborhoods generated
by the pseudometric. However, the neighborhoods can be proper subclasses, so
it won’t be possible to build a class from them. The topological class defined
in [7] allows working in more familiar terms., and with them continuous
curves, their lengths, as well as the notion of intrinsic and strictly
intrinsic generalized pseudometric. In [7] it is proved that the
Gromov–Hausdorff distance is intrinsic, that is, for metric spaces at a finite
distance, it is equal to the infimum of the lengths of the curves connecting
these spaces. The question of whether this distance is strictly intrinsic
remains open. Recall that in the Gromov–Hausdorff space, the distance is a
metric (it is always finite and positively definite) and, as shown in [9],
this metric is strictly intrinsic. It is also well known [5] that the
Gromov–Hausdorff space is complete and separable.

What can we say about the geometry and topology of the Gromov–Hausdorff class?
In this paper we show that the Gromov–Hausdorff class is complete. Then we
split the Gromov–Hausdorff class into “clouds” — maximal proper subclasses,
each of which consists of spaces at a finite distance from each other. A cloud
containing a one-point metric space includes exactly all bounded metric
spaces. All other clouds are made up of unbounded spaces.

To formulate the following result, we introduce some notation. If $X$ is a
metric space, then for a real $\lambda>0$, by $\lambda X$ we denote the metric
space obtained from $X$ by multiplying all distances by $\lambda$. It is well
known that for bounded metric spaces $X$ and $Y$, the Gromov–Hausdorff
distance between $\lambda X$ and $\lambda Y$ equals to the distance between
$X$ and $Y$ multiplying by $\lambda$. In addition, the distance from the one-
point space $\Delta_{1}$ to a space $X$ equals to half the diameter of $X$.
Thus, for $\lambda\to 0+$, all bounded spaces are contracted to $\Delta_{1}$.
It is also clear that multiplication by $\lambda$ transfers the subclass of
bounded spaces into itself. What about this operation for other clouds?

It turns out that everything is much more interesting here. We show by
constructing a concrete example that for elements of some clouds,
multiplication by $\lambda$ can lead to a jump to other clouds. In other
words, for a positive $\lambda$, the distance between a metric space $X$ and
$\lambda X$ may be infinite. Moreover, if for some $\lambda$ the space
$\lambda X$ jumped out of a cloud containing $X$, then, when multiplied by
another $\lambda$, the space $\lambda X$ can return to the original cloud.
There are also such $X$ that never return: when multiplied by any positive
$\lambda\neq 1$, the space $\lambda X$ is at the infinite distance from the
space $X$. Naturally, the problem arises of finding out when the space jumps
out, when it stays, when it returns, and in general how the corresponding
families of $\lambda$ are arranged. In the final part of the paper, we will
present a number of results on this topic using the example of metric spaces
that are subsets of the real line.

Another interesting objects of the research are spaces that remain isometric
to themselves when multiplied by any positive $\lambda$. We called such spaces
$\operatorname{dil}$-invariant. Example of $\operatorname{dil}$-invariant
spaces are a normed space $\mathbb{R}^{n}$, a positive orthant, a bouquet of
$\operatorname{dil}$-invariant spaces, etc. An interesting problem is to
describe both $\operatorname{dil}$-invariant spaces and the spaces at zero
distance from them.

It turns out that if a cloud contains a $\operatorname{dil}$-invariant space,
then the entire cloud is contracted to this space (in the case of
$\mathbb{R}^{n}$ this was discovered by Ilya Belalov). For a cloud containing
spaces jumping out of the cloud when multiplied by some $\lambda>0$, we show
that all other spaces from this cloud behave in the same way. More precisely,
the set of all $\lambda>0$, multiplication by which preserves a space in its
cloud, is the same for all spaces of this cloud.

Thus, in this paper we prove a fundamental fact about the completeness of the
Gromov–Hausdorff class and pose the problem of studying clouds for their
“stability” w.r.t. multiplication by positive $\lambda$. Many interesting
questions remain outside the scope of this work, for example, what could be
the set of those $\lambda$ that preserve the cloud? It is easy to verify that
these sets form multiplicative groups. Can any multiplicative subgroup be so
implemented? Another question: which clouds are stable and which ones are not?
For example, how can one describe the sequences of real numbers that remain in
their clouds? It is clear that all arithmetic progressions are of this kind.
What about increasing geometric progressions? We present a number of results
describing the non-trivial behavior of these progressions.

The authors are grateful to the team of the seminar by A.O. Ivanov and A.A.
Tuzhilin “Theory of extremal networks”, and especially to its co-director
Alexander Ivanov, and also to participant Ilya Belalov, for fruitful
discussions. The study was carried out at Lomonosov Moscow State University,
while A.A.Tuzhilin was supported by Russian Science Foundation, project
21-11-00355.

##  1 Basic definitions and preliminary results

As noted in the introduction, we will study the Gromov–Hausdorff distance on
the proper class of all metric spaces. Let us recall the corresponding
definitions.

In von Neumann-Bernays-Gödel (NGB) set theory, all objects are called
_classes_. Classes are of two types: _sets_ , which are defined as classes
that are elements of other classes, and _proper classes_ , which are not
elemets of any other classes. Many standard operations are defined on the
classes, for example, Cartesian product, mappings, etc. In particular, a
distance function can be specified on each class. We will use the following
terminology:

  * •

a _distance function_ or, in short, a _distance_ on a class $\mathcal{A}$ is
an arbitrary mapping $\rho\colon\mathcal{A}\times\mathcal{A}\to[0,\infty]$,
for which always $\rho(x,x)=0$ and $\rho(x,y)=\rho(y,x)$ (symmetry) hold;

  * •

if the distance satisfies the triangle inequality, i.e., if
$\rho(x,z)\leq\rho(x,y)+\rho(y,z)$ is always satisfied, then $\rho$ is called
_generalized pseudometric_ (the word ”generalized” corresponds to the
possibility of taking the value $\infty$);

  * •

if the positive definiteness condition is additionally satisfied for a
generalized pseudometric, i.e., if $\rho(x,y)=0$ always implies $x=y$, then we
call such $\rho$ _generalized metric_ ;

  * •

finally, if $\rho$ does not take the value $\infty$, then in the above
definitions we will omit the word ‘generalized”, and sometimes, to emphasize
the absence of $\infty$, we call this distance _finite_.

As is customary in metric geometry, instead of $\rho(x,y)$ we write $|xy|$ as
a rule.

As we already noted in the introduction, it is impossible to define a topology
on a proper class, since otherwise the proper class will be an element of this
topology and, therefore, will be a set. Nevertheless, we have proposed a
modified construction, which is also called topology, but already defined on
proper classes.

Let $\mathcal{A}$ be an arbitrary class. Since all its elements are sets, the
cardinality is defined for each of them. For each cardinal number $n$,
consider a subclass $\mathcal{A}_{n}$ consisting of all elements whose
cardinalities do not exceed $n$. Then there is a natural filtering:
$\mathcal{A}_{m}\subset\mathcal{A}_{n}$ for $m\leq n$. This filtering will be
called _filtering by sets_ if all $\mathcal{A}_{n}$ are sets. For a class that
is filtered by sets, we define _topology_ as a mapping $\tau\colon
n\mapsto\tau_{n}$ that maps each cardinal number to a topology on
$\mathcal{A}_{n}$ so that these topologies are _consistent_ , i.e., if $m<n$,
then the topology $\tau_{m}$ is induced from the topology $\tau_{n}$. A class
filtered by sets with the topology defined on it will be called a _topological
class_.

Note that if we take a set of cardinality $n$ as a class $\mathcal{A}$ and
define a topology $\tau\colon m\to\mathcal{A}_{m}$ as described above, then
the topology $\tau_{n}$ on $\mathcal{A}_{n}=\mathcal{A}$ induces topologies on
all $\mathcal{A}_{m}$, $m<n$. On all $\mathcal{A}_{m}$, $m>n$, coinciding in
this case with $\mathcal{A}$, the topology $\tau_{m}$ will coincide with
$\tau_{n}$. Thus, in this case we actually have a standard topology on the set
$\mathcal{A}$ (all other topologies are obtained in the standard way from it),
so our construction can be considered as a generalization of the concept of
topology to proper classes.

Further, if a generalized pseudometric (or metric) is given on a class, then
it turns this class into a topological one in a standard way. The
corresponding topology will be called _pseudometric_ (_metric_).

It follows from the NGB axioms that if $X$ is a set, $\mathcal{A}$ is a class,
and $f\colon X\to\mathcal{A}$ is a mapping, then the image $f(X)$ is a set
that belongs to some $\mathcal{A}_{n}$. Thus, if we take a topological space
as $X$ and a topological class as $\mathcal{A}$, then the _continuity_ of the
mapping $f$ is naturally defined, namely, the mapping $f\colon
X\to\mathcal{A}_{n}$ _is continuous_ if its restriction
$X\mapsto\mathcal{A}_{n}$ is continuous for some and, therefore, for any
cardinal $n$ satisfying the condition $f(X)\subset\mathcal{A}_{n}$.

The latter allows us to introduce in the standard way the definition of a
continuous curve $\gamma$, and with it, in the case when the topology is
generated by a generalized pseudometric, the concepts of the length $|\gamma|$
of a continuous curve $\gamma$, intrinsic pseudometric, as well as strictly
intrinsic pseudometric and geodesic class. Namely, a generalized pseudometric
on a set-filtered class $\mathcal{A}$ will be called _intrinsic_ if for every
$X,Y\in\mathcal{A}$, $|XY|<\infty$, and for any $\varepsilon>0$ there exists a
continuous curve $\gamma$ connecting $X$ and $Y$ such that
$|\gamma|<|XY|+\varepsilon$.

Let us now recall the definitions we need from metric geometry, namely, the
Hausdorff and Gromov–Hausdorff distances.

###  1.1 Hausdorff distance

Let $X$ be an arbitrary metric space, $x\in X$, $r>0$ and $s\geq 0$ be real
numbers. By $U_{r}(x)$ and $B_{s}(x)$ we denote, respectively, the _open_ and
_closed balls_ centered at the point $x$ and with the radii $r$ and $s$. If
$A$ and $B$ are non-empty subsets of $X$, then we put
$|XA|=|AX|=\inf\bigl{\\{}|xa|:a\in A\bigr{\\}}$ and
$|AB|=|BA|=\inf\bigl{\\{}|ab|:a\in A,\,b\in B\bigr{\\}}$. Next, we define the
_open $r$-neighborhood of the set $A$_ by setting $U_{r}(A)=\bigl{\\{}x\in
X:|xA|<r\bigr{\\}}$. Finally, the _Hausdorff distance_ between $A$ and $B$ is
the value

$d_{H}(A,B)=\inf\bigl{\\{}r:A\subset U_{r}(B)\ \&\ U_{r}(A)\supset
B\bigr{\\}}.$

The Hausdorff distance is a generalized pseudometric: it can be infinite, as
in the case of the straight line $\mathbb{R}$ and any of its points, and also
equal to zero between different subsets, for example, between the segment
$[0,1]$ and the interval $(0,1)$. Nevertheless, the following classical result
holds.

######  Theorem 1.1 ([5]).

On the set $\mathcal{H}(X)$ consisting of all non-empty bounded closed subsets
of the metric space $X$, the Hausdorff distance is a metric. Moreover, $X$ and
$\mathcal{H}(X)$ simultaneously possess or not the following properties:
completeness, total boundedness, compactness, bounded compactness.

###  1.2 Gromov–Hausdorff distance

We denote by $\operatorname{\mathcal{VG\\!H}}$ the proper class consisting of
all non-empty metric spaces. On this class, we define a distance function
called the _Gromov–Hausdorff distance_ :

$d_{GH}(X,Y)=\inf\bigl{\\{}d_{H}(X^{\prime},Y^{\prime}):X^{\prime},Y^{\prime}\subset
Z\in\operatorname{\mathcal{VG\\!H}},\,X^{\prime}\approx X,\,Y^{\prime}\approx
Y\bigr{\\}},$

where for the metric spaces $U$ and $V$ the expression $U\approx V$ means that
these spaces are isometric.

The following theorem is well known.

######  Theorem 1.2 ([5]).

The Gromov–Hausdorff distance is a generalized pseudometric vanishing on each
pair of isometric spaces.

This theorem allows us to investigate the Gromov–Hausdorff distance on a “less
wild” proper class $\operatorname{\mathcal{G\\!H}}$ consisting of
representatives of isometry classes of all metric spaces, one from each
isometry class. This is similar to considering all sets up to bijection, that
is, to consider the proper class of all cardinal numbers.

######  Theorem 1.3 ([7]).

The proper class $\operatorname{\mathcal{G\\!H}}$ is filtered by sets, so that
the Gromov–Hausdorff distance turns it into a topological class. Moreover, the
Gromov–Hausdorff distance is an intrinsic generalized pseudometric on
$\operatorname{\mathcal{G\\!H}}$.

The topological class $\operatorname{\mathcal{G\\!H}}$ described in Theorem
1.3 will be called the _Gromov–Hausdorff class_. An important subclass in
$\operatorname{\mathcal{G\\!H}}$ is the proper class which consists of all
bounded metric spaces, i.e., spaces $X$ with finite diameter
$\operatorname{diam}X$. This class will be denoted by $\mathcal{B}$.

Let us describe a few simplest properties of the class
$\operatorname{\mathcal{G\\!H}}$. By
$\Delta_{1}\in\operatorname{\mathcal{G\\!H}}$ we denote the one-point metric
space. Also, for a space $X\in\operatorname{\mathcal{G\\!H}}$ and a real
number $\lambda>0$, we denote by $\lambda X$ the metric space that is obtained
from $X$ by multiplying its all distances by $\lambda$. If $\lambda=0$ and
$\operatorname{diam}X<\infty$, we put $\lambda X=\Delta_{1}$. The
transformation
$H_{\lambda}\colon\operatorname{\mathcal{G\\!H}}\to\operatorname{\mathcal{G\\!H}}$,
$H_{\lambda}\colon X\mapsto\lambda X$ for $\lambda>0$, we call _similarity
with coefficient $\lambda$_.

######  Theorem 1.4 ([5]).

For any $X,Y\in\operatorname{\mathcal{G\\!H}}$,

  1. (1)

$2d_{GH}(\Delta_{1},X)=\dim X$_;_

  2. (2)

$2d_{GH}(X,Y)\leq\max\\{\dim X,\dim Y\\}$_;_

  3. (3)

if the diameter of $X$ or $Y$ is finite, then $\bigl{|}\dim X-\dim
Y\big{|}\leq 2d_{GH}(X,Y)$.

  4. (4)

if the diameter of $X$ is finite, then for any $\lambda\geq 0$ and $\mu\geq 0$
we have $d_{GH}(\lambda X,\mu
X)=\frac{1}{2}|\lambda-\mu|\operatorname{diam}X$, whence it immediately
follows that the curve $\gamma(t):=t\,X$ is shortest between any of its
points, and the length of such a segment of the curve is equal to the distance
between its ends _;_

  5. (5)

for any $\lambda\geq 0$, we have $d_{GH}(\lambda X,\lambda
Y)=\lambda\,d_{GH}(X,Y)$. Note that the only space $X\in\mathcal{B}$ for which
$\lambda X$ is isometric to $X$ for all $\lambda>0$, is the one-point space
$\Delta_{1}$. Thus, the similarity transformation $H_{\lambda}$ is a homothety
of the space $\mathcal{B}$ centered at the one-point metric space.

Figure 1 shows the scheme of the Gromov–Hausdorff class.

Figure 1: Gromov–Hausdorff class, general view.

Consider the relation $\sim_{1}$ on $\operatorname{\mathcal{G\\!H}}$: we say
that $X,Y\in\operatorname{\mathcal{G\\!H}}$ are in relation $\sim_{1}$ if and
only if $d_{GH}(X,Y)=0$. It is easy to see that $\sim_{1}$ is an equivalence.
The equivalence classes of this relation will be called _clouds_. It is clear
that the Gromov–Hausdorff distance between points of the same cloud is finite,
and between points of different clouds is infinite. Thus, the restriction of
the Gromov–Hausdorff distance to the cloud is a (finite) pseudometric.

##  2 Completeness of the Gromov–Hausdorff class

The purpose of this subsection is to prove a fundamental property of the
Gromov–Hausdorff class.

######  Theorem 2.1.

The Gromov–Hausdorff class is complete. In particular, all clouds are
complete.

###### Proof.

Let $X_{1},X_{2},\ldots$ be a fundamental sequence. Without loss of
generality, we will assume that $2d_{GH}(X_{n},X_{n+1})\leq 1/2^{n}$. For each
$n$, choose a correspondence $R_{n}\in\mathcal{R}(X_{n},X_{n=1})$ such that
$\operatorname{dis}R_{n}<1/2^{n}$. We put ${\bar{X}}=\sqcup_{n=1}^{\infty}$,
and for each $x_{1}\in X_{1}$, we call the sequence $x_{1},x_{2}\in{\bar{X}}$
a _thread starting at $x_{1}$_ if for any $n\geq 1$ it holds $x_{n+1}\in
R_{n}(x_{n})$. We denote the set of all such threads by $N({\bar{X}})$. For
threads $\nu=(x_{1},x_{2},\ldots)$ and
$\nu^{\prime}=(x^{\prime}_{1},x^{\prime}_{2},\ldots)$, consider the sequence
of real numbers $|x_{1}x^{\prime}_{1}|,|x_{2}x^{\prime}_{2}|,\ldots$. Since
$\operatorname{dis}R_{n}<1/2^{n}$ then
$\bigl{|}|x_{n}x^{\prime}_{n}|-|x_{n+1}x^{\prime}_{n+1}|\bigr{|}<1/2^{n}$,
whence for any $m\geq 1$ we have

$\bigl{|}|x_{n}x^{\prime}_{n}|-|x_{n+m}x^{\prime}_{n+m}|\bigr{|}=\bigl{|}|x_{n}x^{\prime}_{n}|-|x_{n+1}x^{\prime}_{n+1}|+|x_{n+1}x^{\prime}_{n+1}|-\cdots-|x_{n+m}x^{\prime}_{n+m}|\bigr{|}\\\
\leq\sum_{k=1}^{m}\bigl{|}|x_{n+k-1}x^{\prime}_{n+k-1}|-|x_{n+k}x^{\prime}_{n+k}|\bigr{|}<\sum_{k=1}^{m}\frac{1}{2^{n+k-1}}<\frac{1}{2^{n-1}},$

therefore, the sequence of numbers
$|x_{1}x^{\prime}_{1}|,|x_{2}x^{\prime}_{2}|,\ldots$ is fundamental and,
therefore, there is a limit which we denote by $|\nu\nu^{\prime}|$. It is
clear that these limits define a distance function on $N({\bar{X}})$. Since
for any three threads $\nu=(x_{1},x_{2},\ldots)$,
$\nu^{\prime}=(x^{\prime}_{1},x^{\prime}_{2},\ldots)$,
$\nu^{\prime\prime}=(x^{\prime\prime}_{1},x^{\prime\prime}_{2},\ldots)$, and
for each $n$, the triangle inequalities hold for the points $x_{n}$,
$x^{\prime}_{n}$, $x^{\prime\prime}_{n}$, the same is true for the distances
between $\nu$, $\nu^{\prime}$, and $\nu^{\prime\prime}$. Thus, we have defined
a pseudometric on $N({\bar{X}})$. Having factorized the resulting pseudometric
space with respect to the equivalence relation generated by zero distances, we
obtain a metric space which we denote by $X$. For a thread $\nu\in
N({\bar{X}})$, the class of this equivalence containing $\nu$ we denote by
$[\nu]$.

For each $n$, consider the relation $R^{\prime}_{n}\subset X\times X_{n}$
defined as follows: for each $x$, consider all threads
$\nu=(x_{1},x_{2},\ldots,x_{n},\ldots)\in x$ and put all $(x,x_{n})$ into
$R^{\prime}_{n}$. Since each $x_{n}$ belongs to some thread, then
$R^{\prime}\in\mathcal{R}(X,X_{n})$. Let us estimate the distortion of the
correspondence $R^{\prime}$. Choose arbitrary $x,x^{\prime}\in X$, threads
$\nu=(x_{1},x_{2},\ldots,x_{n},\ldots)\in x$ and
$\nu^{\prime}=(x^{\prime}_{1},x^{\prime}_{2},\ldots,x^{\prime}_{n},\ldots)\in
x^{\prime}$, then $|\nu\nu^{\prime}|=\lim_{k\to\infty}|x_{k}x^{\prime}_{k}|$,
and as shown above,
$\bigl{|}|x_{n}x^{\prime}_{n}|-|x_{n+m}x^{\prime}_{n+m}|\bigr{|}<1/2^{n-1}$,
whence $\bigl{|}|x_{n}x^{\prime}_{n}|-|\nu\nu^{\prime}|\bigr{|}\leq
1/2^{n-1}$. Thus, $\operatorname{dis}R^{\prime}\leq 1/2^{n-1}$, therefore,
$X_{n}\xrightarrow{\operatorname{\mathcal{G\\!H}}}X$. ∎

##  3 Contracted and rain clouds

Starting from this section, we will study how the similarity transformation
changes the Gromov–Hausdorff distance, in particular, whether the metric space
remains in the cloud, or jumps to another one.

######  Corollary 3.1.

Suppose that in a cloud $\mathcal{C}$ there exists $X\in\mathcal{C}$ such that
for some $0<\lambda<1$, the condition $\lambda X\in\mathcal{C}$ is fulfilled.
Then the mapping
$H_{\lambda}\colon\operatorname{\mathcal{G\\!H}}\to\operatorname{\mathcal{G\\!H}}$,
$H_{\lambda}\colon Y\mapsto\lambda Y$, transforms the cloud $\mathcal{C}$ into
itself, and its restriction to $\mathcal{C}$ is a contracting mapping. Thus,
for any $X,Y\in\operatorname{CC}$, there are limits
$X^{\prime}=\lim_{n\to\infty}H_{\lambda}^{n}(X)$,
$Y^{\prime}=\lim_{n\to\infty}H_{\lambda}^{n}(Y)$, with
$d_{GH}(X^{\prime},Y^{\prime})=0$.

###### Proof.

For any $Y\in\mathcal{C}$, we have $d_{GH}(\lambda X,\lambda Y)=\lambda
d_{GH}(X,Y)<\infty$, therefore $\lambda Y\in\mathcal{C}$, and the mapping
$H_{\lambda}$ is contracting. Since the class $\operatorname{\mathcal{G\\!H}}$
is complete by Theorem 2.1, all sequences $H_{\lambda}^{n}(Z)$ converges as
$n\to\infty$, and all their limits are at zero distance from each other (the
uniqueness result on the limit of a contracting mapping in the case of
pseudometric spaces). ∎

For $X$ from a cloud $\mathcal{C}$, we denote by $\Lambda_{X}$ the set of all
$\lambda>0$ for which $\lambda X\in\mathcal{C}$. The following result
immediately follows from the equality $d_{GH}(\lambda X,\lambda Y)=\lambda
d_{GH}(X,Y)$.

######  Proposition 3.2.

The set $\Lambda_{X}$ does not depend on the choice of $X\in\mathcal{C}$.

Proposition 3.2 allows us to give the following definition: for a cloud
$\mathcal{C}$, we put $\Lambda_{\mathcal{C}}=\Lambda_{X}$ for arbitrary
$X\in\mathcal{C}$ and call it the _cloud $\mathcal{C}$ stabilizer_.

######  Remark 3.3.

It is easy to see that the stabilizer $\Lambda_{\mathcal{C}}$ of the cloud
$\mathcal{C}$ is a subgroup in the multiplicative group of positive real
numbers. In addition, Example 4.3 (see below) shows that
$\Lambda_{\mathcal{C}}$ does not preserve the additive structure: the sum of
elements from $\Lambda_{\mathcal{C}}$ may belong not to
$\Lambda_{\mathcal{C}}$.

######  Example 3.4.

Let $\mathcal{C}$ be a cloud containing $\mathbb{R}^{n}$. Then for any
$\lambda>0$, the spaces $\lambda\mathbb{R}^{n}$ and $\mathbb{R}^{n}$ are
isometric. This implies that for any $X\in\mathcal{C}$ and every $\lambda>0$,
we have $d_{GH}(\lambda X,\mathbb{R}^{n})=\lambda\,d_{GH}(X,\mathbb{R}^{n})$,
so $\lambda X\xrightarrow{\operatorname{\mathcal{G\\!H}}}\mathbb{R}^{n}$ as
$\lambda\to 0+$. In particular, $\Lambda_{\mathcal{C}}=(0,\infty)$.

This example was first considered by Ilya Belalov.

######  Definition 3.5.

A metric space $X$ is called _$\operatorname{dil}$ -invariant_ if, for all
$\lambda>0$, the spaces $X$ and $\lambda X$ are isometric. In other words,
$\operatorname{dil}$-invariance means that all similarities of a metric space
are contained in the isometry group $\operatorname{Iso}(X)$ of this space.

######  Problem 3.6.

Describe all $\operatorname{dil}$-invariant metric spaces.

######  Definition 3.7.

If in a cloud $\mathcal{C}$ there is a $Z$ such that for all $X\in\mathcal{C}$
it holds $\lambda X\xrightarrow{\operatorname{\mathcal{G\\!H}}}Z$ as
$\lambda\to 0+$, then such a cloud is called _contractible_.

Similarly with Example 3.4, the following result can be proved.

######  Corollary 3.8.

Each cloud $\mathcal{C}$ containing a $\operatorname{dil}$-invariant space is
contractible. In particular, it holds $\Lambda_{\mathcal{C}}=(0,\infty)$.

The definition of a $\operatorname{dil}$-invariant space can be generalized.

######  Definition 3.9.

We say that a metric space $X$ is a _generalized
$\operatorname{dil}$-invariant_ if for any $\lambda>0$, we have
$\partial_{GH}(X,\lambda X)=0$.

######  Corollary 3.10.

Each cloud $\mathcal{C}$ containing a generalized
$\operatorname{dil}$-invariant space is contractible. In particular, it
satisfies $\Lambda_{\mathcal{C}}=(0,\infty)$.

######  Problem 3.11.

Is it true that each cloud $\mathcal{C}$ with
$\Lambda_{\mathcal{C}}=(0,\infty)$ is contractible?

######  Definition 3.12.

For $A\subset(0,\infty)$, a metric space $X$ is called _$A$ -invariant_ if for
all $\lambda\in A$ it holds $d_{GH}(X,\lambda X)=0$.

######  Corollary 3.13.

Let $\lambda\in(0,1)$ and $A=\\{\lambda^{r}:r\geq 0,\,r\in{\mathbb{Q}}\\}$.
Suppose that there exist a cloud $\mathcal{C}$ and a metric space
$X\in\mathcal{C}$ such that for all $\lambda^{r}\in A$ the spaces
$\lambda^{r}X$ belong to $\mathcal{C}$. Let $Z$ be the limiting space for the
fundamental sequence $X_{n}=H_{\lambda}^{n}(X)$. Then for all $\lambda^{r}\in
A$, the space $Z$ is also a limiting one for the fundamental sequence
$Y_{n}=H_{\lambda^{r}}^{n}(X)=H_{\lambda^{rn}}(X)$. Moreover, $Z$ is
$A$-invariant.

###### Proof.

The sequence $\lambda^{rn}$ contains a subsequence $\lambda^{n_{i}}$,
$n_{i}\in{\mathbb{N}}$. Indeed, if $r=p/q$, $p,q\in{\mathbb{Z}}$, $p\geq 0$,
$q>0$, then for $n=qi$ we get $\lambda^{rn}=\lambda_{pi}$. It remains to set
$n_{i}=pi$. Thus, $d_{GH}(Y_{n_{i}},Z)\to 0$ as $i\to\infty$, whence the
fundamentality of the sequence $Y_{n}$ implies
$Y_{n}\xrightarrow{\operatorname{\mathcal{G\\!H}}}Z$.

On the other hand, for each $r\in A$, the sequence $\lambda^{rn}$ contains a
subsequence $\lambda^{n_{i}}\lambda^{r}$, $n_{i}\in{\mathbb{N}}$. Indeed,
putting $r=p/q$ again and choosing $n$ of the form $1+qi$, we get
$\lambda^{rn}=\lambda^{r+pi}=\lambda^{pi}\lambda^{r}$. It remains to put
$n_{i}=pi$. Thus, $Z$ is the limit of the sequence
$H_{\lambda}^{n_{i}}(\lambda^{r}X)$. However, the $\lambda^{r}Z$ is the limit
of this sequence as well. From the “uniqueness of the limit” it follows that
$d_{GH}(Z,\lambda^{r}Z)=0$. The proof is complete. ∎

######  Definition 3.14.

If $\Lambda_{\mathcal{C}}\neq(0,\infty)$, then we call the cloud $\mathcal{C}$
_rain_ , and each of its elements a _drop_.

##  4 Rain clouds

Let us show that rain clouds exist by constructing a series of examples of
monotonically increasing sequences of real numbers, which, with an induced
metric from the real line, are drops in their corresponding clouds. We start
with a few general results.

######  Lemma 4.1.

Let $X,Y\in\operatorname{\mathcal{G\\!H}}$ be unbounded spaces for which there
exists $R\in\mathcal{R}(X,Y)$ with $\operatorname{dis}R<\infty$. Then for each
$x_{0}\in X$, $y_{0}\in Y$, and $r>0$, there exists $(x,y)\in R$ such that
$x\in X\setminus U_{r}(x_{0})$ and $y\in Y\setminus U_{r}(y_{0})$.

###### Proof.

Suppose the opposite, i.e., for some $x_{0}\in X$ and $y_{0}\in Y$, there
exists $r>0$ such that each $(x,y)\in R$ satisfies
$\min\bigl{\\{}|xx_{0}|,|yy_{0}|\bigr{\\}}<r$. Since the space $Y$ is
unbounded, there exists a sequence $y_{k}\in Y$ such that
$|y_{0}y_{k}|\to\infty$ as $k\to\infty$. Since $R$ is a correspondence, for
every $k$ there exists $x_{k}$ such that $(x_{k},y_{k})\in R$. But then, for
all sufficiently large $k$, we have $|x_{0}x_{k}|<r$. Therefore, choosing such
a $k$, we get

$\operatorname{dis}R\geq\sup_{l}\bigl{|}|x_{k}x_{l}|-|y_{k}y_{l}|\bigr{|}=\infty,$

a contradiction. ∎

######  Theorem 4.2.

Suppose that for unbounded spaces $X,Y\in\operatorname{\mathcal{G\\!H}}$, the
following condition holds _:_ there exist $x_{0}\in X$ and $y_{0}\in Y$ such
that

$\Delta(r)=\inf\Bigl{\\{}\bigl{|}|xx^{\prime}|-|yy^{\prime}|\bigr{|}:x,x^{\prime}\in
X\setminus U_{r}(x_{0}),\ y,y^{\prime}\in Y\setminus U_{r}(y_{0}),\ x\neq
x^{\prime},y\neq y^{\prime}\Bigr{\\}}\to\infty$

for $r\to\infty$. Then for each correspondence $R\in\mathcal{R}$ we have
$\operatorname{dis}R=\infty$, i.e., $d_{GH}(X,Y)=\infty$.

###### Proof.

Suppose the opposite, i.e., let $\operatorname{dis}R<\infty$. Lemma 4.1
implies that for every $r>0$, there exists $(x,y)\in R$ with $|x_{0}x|\geq r$
and $|y_{0}y|\geq r$. Choose $r^{\prime}>r$ such that
$r^{\prime}>\max\bigl{\\{}|x_{0}x|,|y_{0}y|\bigr{\\}}$, then the pair
$(x^{\prime},y^{\prime})\in R$ corresponding $r^{\prime}$ satisfies $x\neq
x^{\prime}$ and $y\neq y^{\prime}$. But then,

$\operatorname{dis}R\geq\bigl{|}|xx^{\prime}|-|yy^{\prime}|\bigr{|}\geq\Delta(r),$

and since $r$ can be chosen arbitrarily large, we get
$\operatorname{dis}R=\infty$, a contradiction. ∎

######  Example 4.3.

Let $p$ be a prime number greater than $2$, and
$X=\\{x_{1}=p,\,x_{2}=p^{2},\,x_{3}=p^{3},\ldots\\}\subset\mathbb{R}$. Put
$Y=2X$ and implement it as a sequence $y_{i}=2x_{i}$. Let us show that
$d_{GH}(X,Y)=\infty$ and thus $X$ is a drop in its cloud $\mathcal{C}$. To do
that, we use Theorem 4.2 to show that $\Delta(r)\to\infty$ as $r\to\infty$.

Choose arbitrary $x,x^{\prime}\in X$, $x\neq x^{\prime}$, and $y,y^{\prime}\in
Y$, $y\neq y^{\prime}$, in such a way that

$\min\bigl{\\{}|x_{1}x|,\,|x_{1}x^{\prime}|,\,|y_{1}y|,\,|y_{1}y^{\prime}|\bigr{\\}}\geq
r.$

Then for some positive integers $m$, $l$, $n$, $k$ we get
$|xx^{\prime}|=p^{m}-p^{l}$, $|yy^{\prime}|=2(p^{n}-p^{k})$. Since $x\neq
x^{\prime}$ and $y\neq y^{\prime}$, then $m>l$ and $n>k$.

Put $s=\min\\{m,l,n,k\\}$, then the condition $r\to\infty$ is equivalent to
$s\to\infty$. Further, let

$\delta=p^{m}-p^{l}-2p^{n}+2p^{k}=p^{s}z.$

Let us show that $z\neq 0$, whence the required will follow, since $s$ can be
chosen arbitrarily large, and $|z|\geq 1$, therefore, $|\delta|\to\infty$ as
$s\to\infty$, and $\Delta(r)\to\infty$ as $r\to\infty$, respectively, due to
the arbitrariness of the choice $x$, $x^{\prime}$, $y$, $y^{\prime}$.

Let $z=0$, then $\Delta=0$, i.e., $p^{l}(p^{m-l}-1)=2p^{k}(p^{n-k}-1)$. Since
$m-l>0$ and $n-k>0$, we get $l=k$, whence the equality $2p^{n-k}-p^{m-l}=1$
follows. Since $n-k>0$ and $m-l>0$, the left side of the equality is a
multiple of $p$, a contradiction.

######  Remark 4.4.

Note that if in the construction from Example 4.3 we multiply the space $X$ by
$p$, then the space will remain in the cloud, since $d_{GH}(X,pX)<\infty$.
Thus, it is not true that the existence of a metric space “jumping out” of its
cloud for some $\lambda$, it follows that for all $\lambda>0$, other than $1$,
the space $\lambda X$ does not lie in the cloud as well.

##  5 An example of studying the structure of a cloud stabilizer

Let $\varphi\colon{\mathbb{N}}\to\mathbb{R}$ be such a monotonically
increasing function of a positive integer argument that $\varphi(n)\geq n$.
For a real number $q>1$, consider the set

(1) $X_{\varphi}=\\{x_{n}=q^{\varphi(n)}:n\in{\mathbb{N}}\\}\subset\mathbb{R}$

with the metric induced from $\mathbb{R}$. We denote the cloud that contains
the space $X_{\varphi}$ by the same symbol.

In the case of the function $\varphi(n)=n$, we denote the space $X_{\varphi}$
by $X_{q}$. Recall that by
$H_{\lambda}\colon\operatorname{\mathcal{G\\!H}}\to\operatorname{\mathcal{G\\!H}}$,
$\lambda>0$, we denoted the similarity transformation $H_{\lambda}\colon
X\mapsto\lambda X$. Sometimes we will refer to the space $X_{q}$ as the space
$\\{x_{n}=q^{n}:n\in{\mathbb{Z}}\\}\subset\mathbb{R}$, which defines the same
cloud but is invariant under homothety $H_{q}$.

######  Corollary 5.1.

If $q\in{\mathbb{N}}$ and $\lambda\in\Lambda_{X_{\varphi}}\cap{\mathbb{Q}}$,
then for any $M$ there exist $n>k>M$ and $m>l>M$ such that

(2)
$\lambda=\frac{q^{\varphi(n)}-q^{\varphi(k)}}{q^{\varphi(m)}-q^{\varphi(l)}}.$

###### Proof.

Let us show that otherwise $d_{GH}(X,\lambda X)=\infty$. To do this, we use
Theorem 4.2 to show that $\Delta(r)\to\infty$ as $r\to\infty$.

Let $\lambda=a_{2}/a_{1}$ for some coprime positive integers $a_{1}$ and
$a_{2}$. Choose arbitrary $x,x^{\prime}\in X$, $x\neq x^{\prime}$, and
$y,y^{\prime}\in Y$, $y\neq y^{\prime}$ such that

$\min\bigl{\\{}|x_{1}x|,\,|x_{1}x^{\prime}|,\,|y_{1}y|,\,|y_{1}y^{\prime}|\bigr{\\}}\geq
r.$

Then for some positive integers $n$, $k$, $m$, $l$ we have
$|xx^{\prime}|=q^{\varphi(n)}-q^{\varphi(k)}$,
$|yy^{\prime}|=\lambda(q^{\varphi(m)}-q^{\varphi(l)})$. Since $x\neq
x^{\prime}$ and $y\neq y^{\prime}$, then $n>k$ and $m>l$.

Let $s=\min\\{n,k,m,l\\}$, then the condition $r\to\infty$ is equivalent to
$s\to\infty$. Further, let

$\delta=q^{\varphi(n)}-q^{\varphi(k)}-\lambda(q^{\varphi(m)}-q^{\varphi(l)})=q^{\varphi(s)}z.$

Suppose that $z\neq 0$. Since $s>M$ can be chosen as large as desired, and the
number $a_{1}z$ is an integer, then $|a_{1}z|\geq 1$. Then $|\delta|\to\infty$
as $s\to\infty$ (since the number $a_{1}$ is fixed) and, respectively,
$\Delta(r)\to\infty$ as $r\to\infty$ due to the arbitrariness of the choice of
$x$, $x^{\prime}$, $y$, $y^{\prime}$.

Therefore, $z=0$, i.e., the number $\lambda$ has the required form. ∎

######  Theorem 5.2.

For a rational number $\lambda>0$ and an integer $q\geq 2$, the similarity
transformation $H_{\lambda}$ keeps the space
$X_{q}=\\{q^{n}:n\in{\mathbb{Z}}\\}$ in the same cloud, i.e.,
$\lambda\in\Lambda_{X_{q}}\cap{\mathbb{Q}}$, if and only if
$\lambda=q^{\alpha}$ for some $\alpha\in{\mathbb{Z}}$.

We split the proof into several elementary steps some of which are undoubtedly
well known and presented here for completeness and consistency.

For a given positive integers $n$, $m$, and an integer $q\geq 2$, we describe
all solutions of the equation

(3) $a_{1}(q^{n}-1)=a_{2}(q^{m}-1)$

in positive integer coprime numbers $a_{1}$, $a_{2}$.

######  Proposition 5.3.

For any integer number $q\geq 2$, we have

(4) $\gcd(q^{n}-1,q^{m}-1)=q^{\gcd(n,m)}-1.$

###### Proof.

We use induction on the number $\max(n,m)$. If $n=m$, then the Equality (4) is
obvious. If $n>m$, then the inductive transition is given by the chain of
equalities

(5) $\gcd(q^{n}-1,q^{m}-1)=\gcd(q^{n}-1-q^{n-m}(q^{m}-1),q^{m}-1)=\\\
=\gcd(q^{n-m}-1,q^{m}-1)=q^{\gcd(n-m,m)}-1=q^{\gcd(n,m)}-1.$

∎

######  Corollary 5.4.

If the positive integers $n$ and $m$ are coprime, then for every integer
$q\geq 2$, the numbers

(6) $q^{n-1}+q^{n-2}+\ldots+q+1\text{ and }q^{m-1}+q^{m-2}+\ldots+q+1\\\ $

are also coprime.

######  Theorem 5.5.

For given positive integers $n$, $m$, and an integer $q\geq 2$, positive
coprime integers $a_{1}$, $a_{2}$ solves the Equation $(\ref{eq:4})$ if and
only if

(7) $\begin{cases}a_{1}=q^{(r_{m}-1)d}+q^{(r_{m}-2)d}+\ldots+q^{d}+1,\\\
a_{2}=q^{(r_{n}-1)d}+q^{(r_{n}-2)d}+\ldots+q^{d}+1,\end{cases}$

where $d=\gcd(n,m)$, $r_{n}=n/d$, and $r_{m}=m/d$.

###### Proof.

Substitute $x=q^{d}$ into the well-known equality

$x^{r}-1=(x-1)(x^{r-1}+\ldots+x+1)$

with $r=n/d$ and $r=m/d$. Thus, according to Equation (4), the Equation (3)
can be rewritten as

(8)
$a_{1}(q^{(r_{n}-1)d}+q^{(r_{n}-2)d}+\ldots+q^{d}+1)=a_{2}(q^{(r_{m}-1)d}+q^{(r_{m}-2)d}+\ldots+q^{d}+1).$

From Corollary 5.4 and the condition $\gcd(a_{1},a_{2})=1$ the required
equalities follow. ∎

######  Proposition 5.6.

For a rational number $\lambda>0$ and an integer $q\geq 2$, if the similarity
transformation $H_{\lambda}$ keeps the space
$X_{q}=\\{q^{n}:n\in{\mathbb{Z}}\\}$ in the same cloud, i.e., if
$\lambda\in\Lambda_{X_{q}}\cap{\mathbb{Q}}$, then

(9)
$\lambda=q^{\alpha}\frac{1+q^{d}+\ldots+q^{(r_{2}-2)d}+q^{(r_{2}-1)d}}{1+q^{d}+\ldots+q^{(r_{1}-2)d}+q^{(r_{1}-1)d}}=q^{\alpha}\frac{q^{r_{2}d}-1}{q^{r_{1}d}-1}$

for some integer $\alpha$, positive integer $d$, and coprime positive integers
$r_{1}$ and $r_{2}$.

###### Proof.

By Corollary 5.1, there exist $n>k$ and $m>l$ such that
$\lambda=\frac{q^{n}-q^{k}}{q^{m}-q^{l}}=q^{k-l}\frac{q^{n-k}-1}{q^{m-l}-1}$.
∎

######  Proposition 5.7.

If the square of the number $\lambda=\frac{q^{n}-1}{q^{m}-1}$ looks like the
right hand side of Equation $(\ref{eq:10})$, then $\lambda=1$.

###### Proof.

Without loss of generality, we can assume that $n>m$. Suppose that there exist
positive integers $k$ and $l$ (with $k>l$) such that

$\lambda^{2}=\frac{(q^{n}-1)^{2}}{(q^{m}-1)^{2}}=\frac{q^{k}-1}{q^{l}-1}.$

Then

$(q^{n}-1)^{2}(q^{l}-1)=(q^{m}-1)^{2}(q^{k}-1),$

i.e.,

$q^{2n+l}-2q^{n+l}+q^{l}-q^{2n}+2q^{n}-1=q^{2m+k}-2q^{m+k}+q^{k}-q^{2m}+2q^{m}-1.$

From divisibility by the number $q$, it follows that $m=l$. After reduction,
the following condition is obtained:

$q^{2n}-2q^{n}+1-q^{2n-m}+2q^{n-m}=q^{m+k}-2q^{k}+q^{k-l}-q^{m}+2$

Under the assumptions $n>m$ and $k>l$, it turns out that $1$ must be divisible
by $q$. ∎

###### Proof of Theorem 5.2.

By Proposition 5.6, all rational numbers in the stabilizer are of the form
(9). Since the stabilizer is a multiplicative group, the square of any number
from the stabilizer has the form (9). It follows from Proposition 5.7 that
$\lambda$ has the required form. ∎

######  Theorem 5.8.

If a function $\varphi\colon{\mathbb{N}}\to\mathbb{R}$ satisfies
$\varphi(n+1)-\varphi(n)\to\infty$ as $n\to\infty$, then for any real numbers
$q>1$ and $\lambda>0$, the similarity transformation $H_{\lambda}$ keeps the
space $X_{\varphi}=\\{q^{\varphi(n)}:n\in{\mathbb{N}}\\}$ in the same cloud if
and only if $\lambda=1$.

###### Proof.

Let $\lambda\neq 1$. Suppose that the number $q$ is an integer, and the number
$\lambda$ is rational. By Corollary 5.1, there exist $n>k>M$ and $m>l>M$ such
that
$\lambda=\frac{q^{\varphi(n)}-q^{\varphi(k)}}{q^{\varphi(m)}-q^{\varphi(l)}}$.

If $n=m$, then

$\lambda=\frac{1-q^{\varphi(k)-\varphi(n)}}{1-q^{\varphi(l)-\varphi(m)}}\stackrel{{\scriptstyle
M\to\infty}}{{\longrightarrow}}1.$

If $M$ is sufficiently large, we get a contradiction with $\lambda\neq 1$.

If $n<m$, then

$\lambda=\frac{1-q^{\varphi(k)-\varphi(n)}}{q^{\varphi(m)-\varphi(n)}-q^{\varphi(l)-\varphi(n)}}\stackrel{{\scriptstyle
M\to\infty}}{{\longrightarrow}}0.$

If $M$ is sufficiently large, we get a contradiction with $\lambda>0$.

If $n>m$, then

$\lambda=\frac{q^{\varphi(n)-\varphi(m)}-q^{\varphi(k)-\varphi(m)}}{1-q^{\varphi(l)-\varphi(m)}}\stackrel{{\scriptstyle
M\to\infty}}{{\longrightarrow}}\infty.$

If $M$ is sufficiently large, we get a contradiction with $\lambda<\infty$.

If $q>1$ and $\lambda>0$ are arbitrary, one needs to return to the proof of
Corollary 5.1. Let us show that for $n>k>M$, $m>l>M$, and $\lambda\neq 1$, the
distortion tend to infinity. Here again there are three cases to consider. If
$n=m$, then

$\delta=q^{\varphi(n)}-q^{\varphi(k)}-\lambda(q^{\varphi(m)}-q^{\varphi(l)})=(1-\lambda)q^{\varphi(n)}\bigl{(}1-\frac{1}{1-\lambda}q^{\varphi(k)-\varphi(n)}+\frac{\lambda}{1-\lambda}q^{\varphi(l)-\varphi(n)}\bigr{)}\to\pm\infty.$

The cases $n<m$ and $n>m$ can be considered in the same way. ∎

######  Corollary 5.9.

If $\varphi(n)=n^{2}$, then for any real numbers $q>1$ and $\lambda>0$, the
similarity transformation $H_{\lambda}$ keeps the space
$X_{\varphi}=\\{q^{n^{2}}:n\in{\mathbb{N}}\\}$ in the same cloud if and only
if $\lambda=1$.

## References

  * [1] Mendelson E. _An Introduction to Mathematical Logic_ (4th ed.), London: Chapman and Hall/CRC, 1997.
  * [2] Edwards D. _The Structure of Superspace. In: Studies in Topology_ , ed. by Stavrakas N.M. and Allen K.R., New York, London, San Francisco, Academic Press, Inc., 1975.
  * [3] Gromov M. _Structures métriques pour les variétés riemanniennes_ , edited by Lafontaine and Pierre Pansu, 1981.
  * [4] Gromov M. _Metric structures for Riemannian and non-Riemannian spaces_ , Birkhäuser (1999). ISBN 0-8176-3898-9 (translation with additional content).
  * [5] Burago D., Burago Yu., Ivanov S. _A Course in Metric Geometry_ , AMS GSM 33, 2001.
  * [6] Herron D.A. _GromovHausdorff Distance for Pointed Metric Spaces_ , J. Anal., 2016, v. 24, N 1, pp 138.
  * [7] Borzov S.I., Ivanov A.O., Tuzhilin A.A. _Extendability of Metric Segments in Gromov-Hausdorff Distance_. 2020, ArXiv e-prints, arXiv:2009.00458.
  * [8] http://dfgm.math.msu.su/files/0students/2021-dip-Borisova.pdf
  * [9] Ivanov A.O., Nikolaeva N.K., Tuzhilin A.A. _The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic_. ArXiv e-prints, arXiv:1504.03830, 2015.
  * [10] Chowdhury S., Memoli F. _Constructing Geodesics on the Space of Compact Metric Spaces_. ArXiv e-prints, arXiv:1603.02385, 2016.
  * [11] Ivanov A.O., Iliadis S., Tuzhilin A.A. _Realizations of Gromov-Hausdorff Distance_. ArXiv e-prints, arXiv:1603.08850, 2016.


# Towards Algorithmic Fidelity: Mental Health Representation across
Demographics in Synthetic vs. Human-generated Data

###### Abstract

Synthetic data generation has the potential to impact applications and domains
with scarce data. However, before such data is used for sensitive tasks such
as mental health, we need an understanding of how different demographics are
represented in it. In our paper, we analyze the potential of producing
synthetic data using GPT-3 by exploring the various stressors it attributes to
different race and gender combinations, to provide insight for future
researchers looking into using LLMs for data generation. Using GPT-3, we
develop HeadRoom, a synthetic dataset of 3,120 posts about depression-
triggering stressors, by controlling for race, gender, and time frame (before
and after COVID-19). Using this dataset, we conduct semantic and lexical
analyses to (1) identify the predominant stressors for each demographic group;
and (2) compare our synthetic data to a human-generated dataset. We present
the procedures to generate queries to develop depression data using GPT-3, and
conduct analyzes to uncover the types of stressors it assigns to demographic
groups, which could be used to test the limitations of LLMs for synthetic data
generation for depression data. Our findings show that synthetic data mimics
some of the human-generated data distribution for the predominant depression
stressors across diverse demographics.

Keywords: Synthetic Data Generation, LLMs, Bias in LLMs, Mental Health
Datasets

Towards Algorithmic Fidelity: Mental Health Representation across Demographics
in Synthetic vs. Human-generated Data

Shinka Mori, Oana Ignat, Andrew Lee, Rada Mihalcea
---
University of Michigan, Ann Arbor
Ann Arbor, MI, USA
{shinkamo, oignat, ajyl<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

The emergence of Large Language Models (LLMs) poses many exciting use cases in
various applications (Bang et al., 2023). In particular, synthetic data
generation (Tang et al., 2023) has great potential to impact domains such as
mental health, where it can scale hard-to-acquire data to improve medical
information extraction (Agrawal et al., 2022), provide clinical decision
support (Shen et al., 2023), and enhance patient-doctor communication
(Kreimeyer et al., 2017).

Prompt Template |  Sample Output
---|---
“I want you to act like a {race} {gender} who is feeling depressed. Write a blog post to describe the main source of stress in your life’ |  .. I’m not good enough, like I’m not doing enough. I’m struggling to make ends meet and I’m constantly worried about money. I’m worried about my family and their safety ...

Table 1:  Example of prompt templates used for HeadRoom, as well as sample
outputs.

However, before using synthetic data, we need to understand the potential
biases across demographics within the underlying models generating such data.
Otherwise, a subsequent model trained on biased synthetic data can have
undesirable consequences, such as misrepresentations of minority voices or,
specifically in mental health, a misdiagnosis Potts et al. (1991); Call and
Shafer (2018).

To address this need, we conduct extensive analyses to understand the
similarities and differences between synthetic and human-generated data. We
focus on mental health data, specifically depression stressors across races
and genders. We study if GPT-3 accurately captures depression stressors across
demographics and if the stressors map closely to those found in human-
generated data. Argyle et al. (2022) coin the term “algorithmic fidelity” to
describe the degree to which models mimic the real-life distributions for a
particular group. Inspired by this, we aim to measure how accurately GPT-3
represents depression stressors for different demographics with the following
research questions:

RQ1:

What are the depression stressors identified by GPT-3 for different
demographic groups and does it capture demographic biases?

RQ2:

How does synthetic data about depression stressors compare to human-generated
data across demographics?

We closely follow the analyses done by Aguirre et al. (2022) on human-
generated data to discover patterns of depression stressors among
demographics. Namely, we generate a similar dataset by prompting GPT-3 to
produce outputs representative of diverse demographics and compare our
findings to theirs.

Our work makes the following contributions. First, we develop and publish
HeadRoom: a syntHEtic dAtaset of Depression-triggering stRessOrs acrOss
deMographics, using GPT-3 while controlling for race, gender, context, and
time phase – before and after COVID-19. Second, we identify the most
predominant depression stressors for each demographic group. Third, we conduct
semantic and syntactic analyses to compare our synthetic data to a human-
generated dataset. Our findings show that GPT-3 exhibits some degree of
“algorithmic fidelity” – the generated data mimics some real-life data
distributions for the most prevalent depression stressors among diverse
demographics.

## 2\. Related Work

#### LLMs for Generating Mental Health Datasets Across Demographics.

Psychological studies show that depression affects racial and gender groups
differently Brody et al. (2018). Despite this, there are still discrepancies
where minority groups are often overlooked for depression diagnoses Stockdale
et al. (2008). While demographic information is a key aspect to consider when
conducting mental health studies, obtaining such data in the mental health
domain is challenging due to safety and privacy regulations Mattern et al.
(2022). As a result, researchers often turn to alternative methods of
obtaining demographic labels, such as using automated classifiers, keywords,
or lists of names Wang and Jurgens (2018). However, as presented in Field et
al. (2021), such methods fail to account for the multidimensionality of race
due to simplifications inherent in classification models: i.e., classifiers
predicting demographics in tweets perform poorly on Asian and Hispanic samples
(Wood-Doughty et al., 2021). Furthermore, commonly used mental health
datasets, such as CLPsych (Coppersmith et al., 2015) and Multitask (Benton et
al., 2017), underrepresent specific demographics such as men and Hispanic
individuals (Aguirre et al., 2021).

An alternative to predicting demographic labels using machine learning is to
_generate_ demographic data using LLMs. Argyle et al. (2022) show that GPT-3
can generate political stances regarding recent elections in the United States
that strongly correlate with real-life voter distributions. Møller et al.
(2023) compare the performance of classifiers trained on human-generated
versus LLM-generated data, demonstrating that classifiers trained on synthetic
data can perform well on tasks such as social dimensions.

Considering the inherent risks of applying these approaches to mental health
tasks, measuring the algorithmic fidelity of LLMs in the mental health domain
is essential. For instance, Lin et al. (2022) demonstrate that LLMs carry
different mental health stigmas for men and women. In our work, we also study
demographic biases in LLMs by generating synthetic data using GPT-3 and
analyzing it against human-generated data.

#### Depression-triggering Stressors Across Demographics Analysis.

Depression stressors can vary greatly depending on the demographic, due to
systemic racism, racial dynamics, gender discrimination, immigration status,
and other factors such as COVID-19 (McKnight-Eily et al., 2021). Specifically,
Loveys et al. (2018) analyze data from self-reported depression users in an
online peer support community.111https://www.7cups.com/ Similar to our work,
Loveys et al. (2018) use Linguistic Inquiry and Word Count (LIWC) Pennebaker
et al. (2007, 2015), a lexical analysis tool, to compare the stressors between
racial groups, and found some critical differences in stressor patterns
between demographics.

## 3\. Datasets

To answer our research questions, we generate data using GPT-3 while
controlling for race, gender, and time (before and after COVID-19), to
simulate human-generated data. We then conduct semantic and lexical analyses
to find patterns in the synthetic data. Next, we compare our findings to those
based on The University of Maryland - OurDataHelps dataset (UMD-ODH) Kelly et
al. (2020, 2021), a demographically diverse human-generated dataset about
depression stressors.

### 3.1. UMD-ODH: Human-generated Data

UMD-ODH (Kelly et al., 2020, 2021) contains open-ended responses from patients
clinically diagnosed with depression and psychosis. Patients were asked:
“Describe the biggest source of stress in your life at the moment. What things
have you done to deal with it?”

Aguirre et al. (2022) further process this data by selecting the survey
responses that have demographic data available, resulting in $2,607$ survey
responses. The resulting demographic information is shown in Table 2.

| Race | Gender | COVID-19
---|---|---|---
Responses | White | Black | Asian | Latinx | Other | Male | Female | Other | Before Pandemic | After Pandemic
UMD-ODH | 1,761 | 221 | 246 | 277 | 102 | 1,857 | 659 | 94 | 890 | 1,717
HeadRoom | 780 | 780 | 780 | 780 | – | 1,500 | 1,500 | – | 1,440 | 1,680

Table 2: Demographic statistics from UMD-ODH and HeadRoom.  Topic |  UMD-ODH |  HeadRoom | Similarity
---|---|---|---
Family |  family, focus, year, planning, friends |  can, stress, deal, lot, person | 0.78
Work |  work, lot, hard, week, balance |  job, stress, work, sourc, life, work | 0.90
Health |  like, feel, surgery, found, productive |  constant, feel, take, health, mental | 0.85
Finance |  money, bills, pay, sleep, lack |  job, struggl, find, make, end | 0.94
Relationship |  day, stressed, relationship, tried, think |  feel, thing, make, depress, like | 0.92
School |  problems, friends, program, plans, dissertation |  asian, succeed, expect, pressur, fall | 0.90
News, Social media |  help, people, social, use, avoid |  like, climat, current, stress, polit | 0.84
Unemployment |  job, new, finding, lost, looking |  look, lost, time, get, month | 0.87
Table 3: The keywords corresponding to each overarching topic in UMD-ODH and
HeadRoom, together with the cosine similarity between the averaged GloVe
embeddings Pennington et al. (2014) of the keywords corresponding to each
topic. A cosine similarity between two sets of random words gives us a
baseline of $0.75$.

### 3.2. HeadRoom: GPT-3 generated Data

We generate our synthetic dataset with GPT-3.222Text-Davinci-003,
https://platform.openai.com/docs/models/gpt-3-5 We use GPT-3 because it is one
of the largest LLMs available, and has been demonstrated to effectively
emulate human texts (Argyle et al., 2022), but our study can be done with any
LLM.333We also attempted to use ChatGPT, but due to its content filters, the
prompt had to be heavily engineered, which may add confounding variables.

#### Prompt Tuning.

To simulate human-generated data, we paraphrase the prompt that Kelly et al.
(2020, 2021) use for their human survey.

We find that we can obtain more detailed responses with additional context in
our prompts. Therefore, we provide additional context such as writing a blog
post, posting on Reddit,444r\Depression or talking to a therapist. We use
three contexts to obtain more diverse responses. For each prompt, we also
specify the user’s gender (women and men), race (Asian, African American,
Hispanic, and White), and context (blog post, Reddit post, and therapy
session). We produce outputs for each race, gender, and context combination.
The prompts and example outputs are displayed in Table 1.

The human-generated data also contains samples collected after the start of
COVID-19, which may affect the stressor patterns. Therefore we also control
for time by indicating the year (2020, 2021) in the prompts while preserving
the data distribution. Prompts that do not indicate the year are assumed to
represent pre-COVID-19 samples.

For the blog post context, we generate $720$ samples, $30$ samples per
demographic group before COVID-19 and $60$ samples after COVID-19. For the
other two contexts, we generate $2,400$ total samples, $150$ samples per
demographic group. The data statistics are summarized in Table 2.

Gender
---
| Category |  Ratio | Category |  Ratio
(a) | Women (+) | Men (-)
| female |  3.95 | male |  -4.06
| i |  2.83 | we |  -1.88
| pro1 |  2.60 | verb |  -1.76
| ppron |  1.32 | tentat |  -1.21
| home |  1.10 | money |  -1.02
Ethnicity
| Category |  Ratio | Category |  Ratio
(b) | Asian (+) | African American (-)
| work |  4.06 | see |  -8.41
| nonflu |  3.11 | percept |  -4.71
| home |  2.31 | health |  -3.88
| reward |  1.57 | and |  -1.76
| achiev |  1.53 | compare |  -1.68
(c) | Asian (+) | White (-)
| leisure |  2.73 | anx |  -2.60
| work |  2.24 | focusfuture |  -1.50
| reward |  2.14 | tentat |  -1.37
| achiev |  1.62 | ingest |  -1.32
| negate |  1.34 | relativ |  -0.99
(d) | Hispanic (+) | White (-)
| home |  7.48 | insight |  -3.36
| leisure |  7.37 | percept |  -3.17
| family |  7.18 | cogproc |  -2.95
| affiliation |  5.30 | see |  -2.62
| social |  2.51 | compare |  -1.83
(e) | African American (+) | White (-)
| see |  5.88 | insight |  -2.28
| bio |  4.01 | adverb |  -2.25
| percept |  3.78 | tentat |  -2.06
| health |  3.40 | you |  -1.80
| body |  2.78 | space |  -1.78
(f) | Hispanic (+) | African American (-)
| home |  7.33 | see |  -8.49
| family |  6.79 | percept |  -6.96
| leisure |  6.72 | bio |  -5.24
| affiliation |  3.67 | health |  -4.67
| social |  3.51 | feel |  -4.40
(g) | Hispanic (+) | Asian (-)
| affiliation |  5.13 | cogproc |  -3.00
| home |  5.04 | i |  -2.75
| leisure |  4.70 | reward |  -2.60
| family |  4.61 | negate |  -2.39
| social |  2.73 | certain |  -2.23

Table 4: Highlights of LIWC log-odds ratio analysis on HeadRoom showing LIWC
categories related to predominant stressors when comparing between genders and
demographic group pairs. For the (+) group, higher score indicates higher
prominence; for the (-) group, lower score indicates higher prominence

## 4\. Dataset Analysis Methods

Following Aguirre et al. (2022), we conduct two analyses on our synthetic
dataset: (1) Semantic analyses using Structural Topic Model (STM), and (2)
Lexical analyses using log-odds-ratio with Latent Dirichlet prior.

#### Semantic Analyses.

STM is a variant of Latent Dirichlet allocation (LDA) that also allows the
addition of covariates, or metadata, to accompany the textual features. Unlike
LDA, which calculates topic prevalence and content from Dirichlet
distributions whose parameters are set in advance, STM uses metadata to find
the topic prevalence and content. Following Aguirre et al. (2022) who
annotated and filtered their topics to $25$, we use gender, race, and time
(before and after COVID-19) and generate $25$ topics. Two annotators labeled
the topics based on their most prevalent keywords, while filtering out unclear
topics. The annotators obtained a Fleiss’ kappa score of $k=0.52$, which shows
a moderate agreement Fleiss (1971, 1973).

We obtain $23$ fine-grained topics that we manually cluster into eleven
overarching topics. The fine-grained and corresponding overarching topics can
be seen in the Appendix (Table 5). Of the eleven overarching topics, eight
match those in the human-generated data. The matching topics and their
keywords are shown in Table 3. Topics from UMD-ODH that did not have a match
with our data include school/ grad school and daily stress. UMD-ODH has two
topics related to school – the first relates to school in general, and the
second relates to graduate school. While our dataset has a topic for general
stress, we concluded that none of the keywords are similar enough to be
considered a match.555feeling stuck, staying strong, uncertainty, comparing to
others, helplessness, stress and anxiety, loneliness, perfectionism

The four overarching topics that appear in the synthetic data and not in the
human-generated data are: general stress, racism and police violence,
immigration status and pandemic. We conduct a pairwise analysis between each
gender and race pair using these topics. Effectively, to find the difference
between topic proportions for each demographic pair, we estimate a regression
to find the topic proportion with the added covariate information. This is
then used to extract the prevalence of a topic (topic distribution) for each
demographic pair. We show and discuss the difference in the topic prevalence
for each demographic pair in Figure 2 and Figure 3.

#### Lexical Analyses.

LIWC Pennebaker et al. (2007, 2015) includes dictionaries of English words
related to human cognitive processes. Specifically, we use the LIWC 2015
dictionary, which contains $6,400$ word stems. Each word stem is assigned to
multiple categories, e.g., father is assigned to: male, family and social.

Aguirre et al. (2022) apply log-odds-ratio with Latent Dirichlet prior, based
on the work of Monroe et al. (2017), which aims to capture how a demographic
group uses a specific LIWC category compared to another demographic group. For
example, to compare the proportions in which one group uses the negemo
(negative emotion) LIWC category compared to another group, we calculate the
log-odds-ratio to get the odds of negemo being used in the first group
compared to the latter. To calculate the Dirichlet prior, we use the LIWC
category counts in the CLPsych dataset (Coppersmith et al., 2015). Note that
we do not normalize the results with a control text unrelated to depression,
to preserve comparison fidelity with Aguirre et al. (2022), who also do not
normalize.

We show the top five words that have a high log-odds-ratio in Table 4 and
highlight the LIWC categories also present in the pairwise lexical analysis
from Aguirre et al. (2022) in the Appendix Tables 6, 7, 8, 9, 10, 11, and 12.
Insights from the data analysis are presented in Section 5.

## 5\. Research Questions

#### RQ1. What are the depression stressors identified by GPT-3 for different
demographic groups, and do they capture demographic biases?

The topic proportion between different demographics, and lexical analyses
indicate demographic differences regarding stressors. Refer to Figure 2,
Figure 3 and Table 4 for the figures.

#### Gender.

Between genders, women have more mentions of first-person pronouns (pro1 and
ppron) (Table 4 (a)). Also, we find the following prevalent stressors: health,
news and social media, news and politics, family, and relationship. See Figure
2 (g), Figure 3 (g).

In contrast, men tend to mention stressors regarding finances and
unemployment, and school more than women. Furthermore, topics regarding racism
and police brutality are much more prominent in men than women.

Both women and men mention stressors related to work, but for different
reasons: women about work2/ work-pressure and men about work1/ work-fatigue.
The two types of work-related stressors are defined in Appendix Table 5.

We acknowledge that we are excluding other gender identities by only comparing
between two genders, women and men. We take this decision because the
comparison data used in Aguirre et al. (2021) is primarily from binary
genders, women, men, and very few from other.

#### Race.

We conduct a pairwise analysis for each race group.

####  African American.

The African American group tends to mention words related to health, body,
perception and family (LIWC categories: bio, health, body, percept, and see).
Topics relating to racism and police brutality are also more likely compared
to other groups. See Table 4 (b, e, f), Figure 2 (a, d, e), and Figure 3 (a,
d, e)

#### Asian.

For the Asian group, the topics perfectionism and comparing to others are
significant stressors. The Asian group also tend to be more concerned with
work, school and reward (LIWC categories: work, reward and achiev). See Table
4 (b, c, g), Figure 2 (a, b, c), and Figure 3 (a, b, c).

#### Hispanic.

The Hispanic group has more stressors related to immigration. Other stressors
include family and social interactions, while other prominent topics include
finances, news, work. See Table 4 (d, f, g), Figure 2 (b, e, f), and Figure 3
(b, e, f).

#### White.

In the White group, the most prominent stressors are: general stress, news and
social media, relationships and uncertainty. See Table 4 (c, d, e), Figure 2
(c, d, f), and Figure 3 (c, d, f).

Figure 1: Topic Modeling: topic proportions between race and gender
intersectionality – African American women vs. African American men. The bars
represent confidence intervals. The closer to the graph extremities, the more
prevalent the topics are for the corresponding demographics

#### Race and Gender Intersectionality.

We also analyze the intersectionality of race and gender in HeadRoom, and
provide an excerpt of the experiment to demonstrate how it could be used to
study the data further. Analyzing all possible demographic combinations would
be too expansive, hence we only provide an excerpt to demonstrate its use
case. Focusing on only one demographic category, such as race or gender, can
overlook the fine-grained inequalities in demographic groups. Field et al.
(2021) give the example that only looking at African American Group emphasizes
the more gender-privileged group (African American men), and similarly, only
looking at gender may lead to over-representing the race-privileged group
(White women). We fit an STM model on the race-gender metadata to find
stressor patterns comparing African American women to African American men. In
Figure 1, we show that police violence and racism are more prevalent stressors
for African American men. African American women are more concerned about
work, pandemic, news and politics, general stress, and relationship and
family.

In future work, if we can access a demographically labeled depression dataset,
we can compare the intersectional stressor patterns to real-life data, as the
data from Aguirre et al. (2022) does not include analyses on
intersectionality.

|
---|---
(a) Asian vs. African American | (b) Asian vs. Hispanic
|
(c) Asian vs. White | (d) African American vs. White
|
(e) Hispanic vs. African American | (f) Hispanic vs. White
(g) Women vs. Men
Figure 2:  Topic Modeling: topic proportion between different demographics, as
detected in GPT-generated data and in real-life data. Colors represent
different races and genders: Men – purple, Women – orange, Asian – magenta,
African American – green, Hispanic – blue, and White – red. The bars represent
confidence intervals. The closer to the graph extremities, the more prevalent
the topics for the corresponding demographics. For example, graph (a) Asian
vs. African American shows that stressors such as work1/ work-fatigue, work2/
work-pressure and school are more prevalent for Asian than for African
American. Best viewed in color.

|
---|---
(a) Asian vs. African American | (b) Asian vs. Hispanic
|
(c) Asian vs. White | (d) African American vs. White
|
(e) Hispanic vs. African American | (f) Hispanic vs. White
(g) Women vs. Men

Figure 3: Topic Modeling: topic proportion between different demographics, as
detected in GPT-generated data and not in real-life data. Colors represent
different races and genders: Men – purple, Women – orange, Asian – magenta,
African American – green, Hispanic – blue, and White – red. The bars represent
confidence intervals. The closer to the graph extremities, the more prevalent
the topics for the corresponding demographics. Best viewed in color.

#### RQ2. How does synthetic data about depression stressors compare to human-
generated data across demographics?

We compare the analysis findings from our synthetic dataset with the findings
from UMD-ODH Aguirre et al. (2022): (1) between the keywords extracted from
the aggregated data, not split by demographics,666We compute over the
aggregated data as we could not obtain keywords split by demographic from
Aguirre et al. (2022). and (2) between the stressors obtained for each
demographic group. When comparing the stressors across demographics, we also
compare them with other findings related to stressor patterns (McKnight-Eily
et al., 2021; Aguirre et al., 2022; Loveys et al., 2018). Our analysis results
as depicted in Figure 2, and Table 4 show that the most prevalent depression
stressors across demographics are comparable between the human-generated and
the synthetic datasets. At the same time, GPT-3 also identifies other
stressors not present in the real-life data, as shown in Figure 3.

#### Topic Similarity.

From Aguirre et al. (2022), we obtain the top 30 most prevalent keywords for
each topic from UMD-ODH. We then compare them with the keywords from our
topics to measure how closely they match each other.

For each topic pair, we convert all keywords in each topic into word
embeddings using GloVe Pennington et al. (2014).777We use glove.6B.300d from
https://nlp.stanford.edu/projects/glove/ We then average the embeddings within
each topic, and calculate the cosine similarity of the averaged embeddings.
Topics with one-to-many matches (e.g., work1/ work-fatigue and work2/ work-
pressure) are consolidated into one topic. Table 3 shows the cosine similarity
scores between each topic pair.

#### Gender.

Comparing gender-related stress patterns, the findings in Aguirre et al.
(2022) show that stressors related to finances, relationships, and health are
more prevalent for women than men. In contrast, stressors about social
interactions (LIWC categories: home, leisure, social, affiliation, we and
family) are more prevalent in men.

Our findings largely support this pattern and show that the prevalence of
health and relationships stressors are more dominant in women.

However, different from Aguirre et al. (2022), we find that in HeadRoom,
stressors related to finance are more dominant in men than in women (LIWC
category: money). See Table 4 (a), Figure 2 (g), and Figure 3 (g).

#### African American Group.

Real-life depression patterns indicate that African Americans are more likely
to discuss health and use more social terms compared to other groups Loveys et
al. (2018). Our synthetic data also supports this: The stressor health is more
prevalent for African American groups than for Asian, White, or Hispanic
groups (LIWC categories: health, body, and bio).

In our synthetic data, we also find that stressors for police brutality,
police violence, and racism are more prominent in African American groups
despite these topics not being present in Aguirre et al. (2022). However,
Alang et al. (2021) showed that hostile police encounters significantly affect
African American and Hispanic individuals and are associated with depressed
mood and anxiety. See Table 4 (b, e, f), Figure 2 (a, d, e), and Figure 3 (a,
d, e).

#### Asian Group.

In our synthetic data, the Asian Group demonstrates a more substantial
prevalence of school stressors, matching prior findings from Aguirre et al.
(2021); Loveys et al. (2018). Surprisingly, topics relating to the impacts of
COVID-19 are more commonly associated with the Asian group, despite prior
findings showing that Hispanic groups were severely affected by it due to lack
of housing and basic needs (McKnight-Eily et al., 2021). See Table 4 (b, c,
g), and Figure 2 (a, b, c).

#### Hispanic Group.

Aguirre et al. (2022); McKnight-Eily et al. (2021) showed that stressors
education, finance, government, and family are prevalent in the Hispanic
group. These findings align with ours: finance, school, family, and politics
are more prevalent for the Hispanic group than other groups

However, Loveys et al. (2018) found that the Hispanic group tends to make
fewer mentions of social terms than African Americans, which contradicts our
findings. We find that LIWC categories social and affiliation are more common
in Hispanic groups. See Table 4 (d, f, g), Figure 2 (b, e, f), and Figure 3
(b, e, f).

#### White Group.

Similar to Aguirre et al. (2022) and McKnight-Eily et al. (2021), we find that
for the White Group, the finance stressor is less prevalent than in other
racial groups (Figure 2 (c, d, f)).

Different from previous findings (McKnight-Eily et al., 2021), when comparing
White and African American groups, we find that family stressors are more
prevalent in the African American group (see Figure 2 (d)).

## 6\. Conclusion

In this paper, we developed a procedure to produce depression data using
GPT-3, which could be applied to other LLMs to test their capability for
creating synthetic mental health data. We perform semantic and lexical
analyses on this dataset to understand how GPT-3 represents depression
stressors across demographics. Our findings show the differences in the types
of depression stressors GPT-3 attributes to different demographics, and that
some prominent stressors across demographics are similar to those in real-life
data from UMD-ODH. Our synthetic data and code is for research purposes only
and is made available at
https://github.com/MichiganNLP/depression_synthetic_data.

## 7\. Acknowledgments

We thank the anonymous reviewers for their constructive feedback. This project
was partially funded by a National Science Foundation award (#2306372). Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
the National Science Foundation. We thank Philip Resnnik and Carlos Aguirre
for providing us with their topic model generation code as well as the data.

## Ethical Statement and Limitations

### 7.1. Ethical Statement

We clarify that the intent of our research nor our dataset is not a proxy for
creating mental health datasets. We see our paper as a way to discover the
biases that LLMs have for different demographics and compare them with
available human data. We do not believe this data should be used for
supplementing current human data because it can enforce biases. Instead, we
propose to use our data for research, to investigate the biases of current
LLMs in mental health and how they compare to human data. Our dataset is
created using only GPT-3, and according to the IRB of our institution, does
not classify as a human subjects research. It is also difficult to explain why
GPT-3, or LLMs in general, speculate these stressors, and lack of
explainability of the outputs should be considered when following our methods.
However, despite the lack of explainability, the recent evolution in the
quality of LLM output is driving researchers to consider its application in
synthetic data generation such as hate-speech dataMøller et al. (2023).
Creating procedures to analyze the algorithmic fidelity of these datasets
encourage researchers to use LLMs for synthetic data generation with caution
by developing a series of methods to understand its underlying behaviors and
potential risks.

### Limitations

#### Gender Representation.

We are aware that the gender and race categories we explored are exclusionary
and do not capture the full spectrum of gender identity, sexuality, race, and
ethnicity; our choice in race and gender groups were predominantly decided by
the availability of existing, human-generated datasets.

#### Location Representation.

The authors who collected the UMD-ODH dataset did not mention the location of
the patients, so we also do not mention it in the GPT-3 prompts Kelly et al.
(2021, 2020). The data it generates is probably not comprehensive of the whole
world, and the findings do not represent all cultures.

#### Sensitive Information.

Across all groups, we note that mentions of suicide or self-harm are not
included in our synthetic data. At the same time, they tend to be mentioned in
real-life depression texts (Aguirre et al., 2022). This difference may be a
result of model restrictions.

#### Dataset Size.

The size of the dataset is based on the UMD-ODH dataset used by Aguirre et al.
(2022) which consisted of 2607 samples; we also keep our synthetic dataset
size small while balancing for demographic groups to conduct a fair comparison
to their results.

#### Using Real-life Depression Data.

Due to the difficulty of obtaining demographically-labeled depression
datasets, we could not conduct fine-grained analyses between our data and
human-generated depression data. While we conduct some quantitative analyses
based on the topic keywords provided by the authors of Aguirre et al. (2022),
having access to a human-generated dataset would have allowed us to obtain
more detailed observations. We also produced relatively short samples, and we
do not know whether these stressor patterns hold for longer text sequences.

#### Model Variability.

At the moment, OpenAI does not mention updating text-davinci-003; however, we
do not know if this will remain true. Possible changes to the model may alter
our findings. Additionally, the model is only trained with data up to June
2021, and cannot predict relevant stressors beyond that time frame. The
prompts used here are flexible in that the time context can be replaced easily
to target a specific time frame, and can be used with other LLMs that has
similar capabilities. One could use another LLM model using our prompt to
explore its potential to be applied in depression analysis.

## References

  * Agrawal et al. (2022) Monica Agrawal, Stefan Hegselmann, Hunter Lang, Yoon Kim, and David Sontag. 2022\.  Large language models are few-shot clinical information extractors.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 1998–2022, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Aguirre et al. (2022) Carlos Aguirre, Mark Dredze, and Philip Resnik. 2022.  Using Open-Ended Stressor Responses to Predict Depressive Symptoms across Demographics.  ArXiv:2211.07932 [cs].
  * Aguirre et al. (2021) Carlos Aguirre, Keith Harrigian, and Mark Dredze. 2021.  Gender and Racial Fairness in Depression Research using Social Media.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 2932–2949, Online. Association for Computational Linguistics.
  * Alang et al. (2021) Sirry Alang, Donna McAlpine, and Malcolm McClain. 2021.  Police Encounters as Stressors: Associations with Depression and Anxiety across Race.  _Socius_ , 7:2378023121998128.  Publisher: SAGE Publications.
  * Argyle et al. (2022) Lisa P. Argyle, Ethan C. Busby, Nancy Fulda, Joshua Gubler, Christopher Rytting, and David Wingate. 2022.  Out of One, Many: Using Language Models to Simulate Human Samples.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 819–862.  ArXiv:2209.06899 [cs].
  * Bang et al. (2023) Yejin Bang, Samuel Cahyawijaya, Nayeon Lee, Wenliang Dai, Dan Su, Bryan Wilie, Holy Lovenia, Ziwei Ji, Tiezheng Yu, Willy Chung, Quyet V Do, Yan Xu, and Pascale Fung. 2023.  A multitask, multilingual, multimodal evaluation of ChatGPT on reasoning, hallucination, and interactivity.
  * Benton et al. (2017) Adrian Benton, Margaret Mitchell, and Dirk Hovy. 2017.  Multi-Task Learning for Mental Health using Social Media Text.  ArXiv:1712.03538 [cs].
  * Brody et al. (2018) Debra J. Brody, Laura A. Pratt, and Jeffery P. Hughes. 2018.  Prevalence of depression among adults aged 20 and over: United states, 2013-2016.  _NCHS data brief_ , (303):1–8.
  * Call and Shafer (2018) Jarrod B. Call and Kevin Shafer. 2018.  Gendered manifestations of depression and help seeking among men.  _American Journal of Men’s Health_ , 12(1):41–51.
  * Coppersmith et al. (2015) Glen Coppersmith, Mark Dredze, Craig Harman, Kristy Hollingshead, and Margaret Mitchell. 2015.  CLPsych 2015 Shared Task: Depression and PTSD on Twitter.  In _Proceedings of the 2nd Workshop on Computational Linguistics and Clinical Psychology: From Linguistic Signal to Clinical Reality_ , pages 31–39, Denver, Colorado. Association for Computational Linguistics.
  * Field et al. (2021) Anjalie Field, Su Lin Blodgett, Zeerak Waseem, and Yulia Tsvetkov. 2021.  A Survey of Race, Racism, and Anti-Racism in NLP.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 1905–1925, Online. Association for Computational Linguistics.
  * Fleiss (1971) Joseph L. Fleiss. 1971.  Measuring nominal scale agreement among many raters.  _Psychological Bulletin_ , 76:378–382.
  * Fleiss (1973) Joseph L. Fleiss. 1973.  Statistical methods for rates and proportions.
  * Kelly et al. (2021) Deanna L. Kelly, Max Spaderna, Vedrana Hodzic, Glen Coppersmith, Shuo Chen, and Philip Resnik. 2021.  Can language use in social media help in the treatment of severe mental illness? _Current research in psychiatry_ , 1(1):1–4.
  * Kelly et al. (2020) Deanna L. Kelly, Max Spaderna, Vedrana Hodzic, Suraj Nair, Christopher Kitchen, Anne E. Werkheiser, Megan M. Powell, Fang Liu, Glen Coppersmith, Shuo Chen, and Philip Resnik. 2020.  Blinded Clinical Ratings of Social Media Data are Correlated with In-Person Clinical Ratings in Participants Diagnosed with Either Depression, Schizophrenia, or Healthy Controls.  _Psychiatry Research_ , 294:113496.
  * Kreimeyer et al. (2017) Kory Kreimeyer, Matthew Foster, Abhishek Pandey, Nina Arya, Gwendolyn Halford, Sandra F. Jones, Richard Forshee, Mark Walderhaug, and Taxiarchis Botsis. 2017\.  Natural language processing systems for capturing and standardizing unstructured clinical information: A systematic review.  _Journal of Biomedical Informatics_ , 73:14–29.
  * Lin et al. (2022) Inna Lin, Lucille Njoo, Anjalie Field, Ashish Sharma, Katharina Reinecke, Tim Althoff, and Yulia Tsvetkov. 2022.  Gendered Mental Health Stigma in Masked Language Models.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 2152–2170, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Loveys et al. (2018) Kate Loveys, Jonathan Torrez, Alex Fine, Glen Moriarty, and Glen Coppersmith. 2018\.  Cross-cultural differences in language markers of depression online.  In _Proceedings of the Fifth Workshop on Computational Linguistics and Clinical Psychology: From Keyboard to Clinic_ , pages 78–87, New Orleans, LA. Association for Computational Linguistics.
  * Mattern et al. (2022) Justus Mattern, Zhijing Jin, Benjamin Weggenmann, Bernhard Schoelkopf, and Mrinmaya Sachan. 2022.  Differentially private language models for secure data sharing.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 4860–4873, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * McKnight-Eily et al. (2021) Lela R. McKnight-Eily, Catherine A. Okoro, Tara W. Strine, Jorge Verlenden, NaTasha D. Hollis, Rashid Njai, Elizabeth W. Mitchell, Amy Board, Richard Puddy, and Craig Thomas. 2021.  Racial and Ethnic Disparities in the Prevalence of Stress and Worry, Mental Health Conditions, and Increased Substance Use Among Adults During the COVID-19 Pandemic — United States, April and May 2020.  _Morbidity and Mortality Weekly Report_ , 70(5):162–166.
  * Monroe et al. (2017) Burt L. Monroe, Michael P. Colaresi, and Kevin M. Quinn. 2017.  Fightin’ Words: Lexical Feature Selection and Evaluation for Identifying the Content of Political Conflict.  _Political Analysis_ , 16(4):372–403.  Publisher: Cambridge University Press.
  * Møller et al. (2023) Anders Giovanni Møller, Jacob Aarup Dalsgaard, Arianna Pera, and Luca Maria Aiello. 2023.  Is a prompt and a few samples all you need? Using GPT-4 for data augmentation in low-resource classification tasks.  ArXiv:2304.13861 [physics].
  * Pennebaker et al. (2007) James W. Pennebaker, Roger John Booth, and Martha E. Francis. 2007.  Linguistic inquiry and word count (LIWC2007).
  * Pennebaker et al. (2015) James W. Pennebaker, Ryan L. Boyd, Kayla Jordan, and Kate G. Blackburn. 2015.  The development and psychometric properties of LIWC2015.
  * Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014.  GloVe: Global vectors for word representation.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1532–1543, Doha, Qatar. Association for Computational Linguistics.
  * Potts et al. (1991) Marilyn K. Potts, M. Audrey Burnam, and Kenneth B. Wells. 1991.  Gender differences in depression detection: A comparison of clinician diagnosis and standardized assessment.  _Psychological Assessment: A Journal of Consulting and Clinical Psychology_ , 3:609–615.
  * Shen et al. (2023) Yiqiu Shen, Laura Heacock, Jonathan Elias, Keith D. Hentel, Beatriu Reig, George Shih, and Linda Moy. 2023.  ChatGPT and Other Large Language Models Are Double-edged Swords.  _Radiology_ , 307(2):e230163.  Publisher: Radiological Society of North America.
  * Stockdale et al. (2008) Susan E. Stockdale, Isabel T. Lagomasino, Juned Siddique, Thomas McGuire, and Jeanne Miranda. 2008.  Racial and ethnic disparities in detection and treatment of depression and anxiety among psychiatric and primary health care visits, 1995-2005.  _Medical Care_ , 46(7):668–677.
  * Tang et al. (2023) Ruixiang Tang, Xiaotian Han, Xiaoqian Jiang, and Xia Hu. 2023.  Does synthetic data generation of LLMs help clinical text mining? ArXiv:2303.04360 [cs].
  * Wang and Jurgens (2018) Zijian Wang and David Jurgens. 2018.  It’s going to be okay: Measuring Access to Support in Online Communities.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 33–45, Brussels, Belgium. Association for Computational Linguistics.
  * Wood-Doughty et al. (2021) Zach Wood-Doughty, Paiheng Xu, Xiao Liu, and Mark Dredze. 2021.  Using Noisy Self-Reports to Predict Twitter User Demographics.  In _Proceedings of the Ninth International Workshop on Natural Language Processing for Social Media_ , pages 123–137, Online. Association for Computational Linguistics.

##  Appendix A Appendix

Overarching Topics | Fine-grained Topic
---|---
Work | work-fatigue (work 1)
work-pressure (work 2)
Racism/ police brutality | Fear of police and violence
Racism
Police brutality
General stress | Feeling stuck
Staying strong
Uncertainty
Comparing to others
Helplessness
Stress and anxiety
Loneliness
Perfectionism
Immigration status | Fear of deportation
Life as an immigrant
News | News and social media
Politics and economy
Finances | Finances and unemployment
Pandemic | Pandemic
Family | Family
Relationships | Relationships
Health | Health
School | School
Table 5: All topics from our synthetic data. The overarching topics that also
match the topics in the UMD-ODH data are highlighted in bold. Gender
---
Women(+) | Men(-)
category | ratio | category | ratio
female | 3.95 | male | -4.06
adverb | 3.16 | see | -2.45
i | 2.83 | we | -1.88
pro1 | 2.56 | verb | -1.76
feel | 1.81 | ipron | -1.70
anx | 1.52 | auxverb | -1.35
ppron | 1.32 | tentat | -1.21
affect | 1.27 | body | -1.21
posemo | 1.26 | article | -1.05
home | 1.10 | money | -1.02
insight | 1.06 | interrog | -1.01
leisure | 1.04 | health | -0.95
sad | 1.02 | compare | -0.89
friend | 1.00 | focuspast | -0.86
conj | 0.99 | discrep | -0.84
Table 6: Lexical analysis of our data between Women and Men. LIWC categories
that also matches the topics in the UMD-ODH data are highlighted in bold
Ethnicity
---
Asian(+) | White
category | ratio | category | ratio
leisure | 2.73 | anx | -2.60
family | 2.62 | adverb | -2.57
certain | 2.60 | see | -2.54
pro1 | 2.57 | motion | -1.84
home | 2.50 | negemo | -1.55
work | 2.24 | focusfuture | -1.49
i | 2.20 | interrog | -1.44
reward | 2.14 | tentat | -1.37
achiev | 1.62 | ingest | -1.32
drives | 1.59 | insight | -1.16
posemo | 1.47 | space | -1.02
negate | 1.34 | relativ | -0.99
auxverb | 1.14 | anger | -0.98
focuspast | 1.09 | percept | -0.92
nonflu | 1.08 | adj | -0.86
Table 7: Lexical analysis of our data between Asian and White group. LIWC categories that also matches the topics in the UMD-ODH data are highlighted in bold Hispanic(+) | White(-)
---|---
category | ratio | category | ratio
home | 7.48 | insight | -3.36
leisure | 7.37 | percept | -3.17
family | 7.18 | cogproc | -2.95
affiliation | 5.30 | see | -2.62
we | 4.36 | feel | -2.62
drives | 2.63 | tentat | -2.39
social | 2.51 | compare | -1.83
focuspast | 2.28 | differ | -1.74
money | 2.25 | ipron | -1.48
anx | 2.01 | health | -1.28
achiev | 1.75 | bio | -1.23
auxverb | 1.53 | space | -1.22
cause | 1.44 | power | -1.12
number | 1.30 | prep | -1.10
pro1 | 1.07 | negate | -1.05
Table 8: Lexical analysis of our data between Hispanic and White group. LIWC categories that also matches the topics in the UMD-ODH data are highlighted in bold African American(+) | White(-)
---|---
category | ratio | category | ratio
see | 5.88 | insight | -2.28
bio | 4.01 | adverb | -2.25
percept | 3.78 | tentat | -2.07
certain | 3.74 | nonflu | -2.04
we | 3.68 | work | -1.81
number | 3.44 | informal | -1.81
health | 3.40 | you | -1.80
body | 2.78 | space | -1.78
time | 2.18 | ppron | -1.67
adj | 1.86 | power | -1.55
feel | 1.77 | differ | -1.45
compare | 1.75 | cogproc | -1.41
prep | 1.66 | shehe | -1.26
affiliation | 1.66 | discrep | -1.25
money | 1.62 | focusfuture | -1.18
Table 9: Lexical analysis of our data between African American and White group. LIWC categories that also matches the topics in the UMD-ODH data are highlighted in bold Hispanic(+) | African American(-)
---|---
category | ratio | category | ratio
home | 7.33 | see | -8.49
family | 6.79 | percept | -6.96
leisure | 6.72 | bio | -5.24
affiliation | 3.67 | health | -4.68
social | 3.51 | feel | -4.40
anx | 2.88 | compare | -3.59
focuspast | 2.81 | certain | -3.36
ppron | 2.47 | prep | -2.77
work | 2.05 | body | -2.74
you | 1.85 | adj | -2.56
drives | 1.76 | number | -2.14
focusfuture | 1.74 | ipron | -2.01
achiev | 1.66 | cogproc | -1.56
nonflu | 1.58 | time | -1.50
informal | 1.47 | quant | -1.17
Table 10: Lexical analysis of our data between Hispanic and African American group. LIWC categories that also matches the topics in the UMD-ODH data are highlighted in bold Hispanic(+) | Asian(-)
---|---
category | ratio | category | ratio
affiliation | 5.13 | cogproc | -3.00
we | 5.06 | feel | -2.99
home | 5.04 | i | -2.75
leisure | 4.70 | reward | -2.60
family | 4.61 | negate | -2.39
anx | 4.61 | percept | -2.26
social | 2.73 | certain | -2.23
negemo | 2.44 | insight | -2.21
focusfuture | 2.05 | work | -2.00
adverb | 1.69 | posemo | -1.94
money | 1.57 | compare | -1.91
motion | 1.22 | nonflu | -1.53
interrog | 1.19 | pro1 | -1.50
focuspast | 1.19 | power | -1.34
verb | 1.14 | differ | -1.09
Table 11: Lexical analysis of our data between Hispanic and Asian group. LIWC categories that also matches the topics in the UMD-ODH data are highlighted in bold Asian (+) | African American (-)
---|---
category | ratio | category | ratio
work | 4.06 | see | -8.41
i | 3.17 | bio | -4.79
nonflu | 3.11 | percept | -4.71
ppron | 2.64 | we | -4.38
home | 2.31 | health | -3.89
family | 2.19 | body | -3.16
leisure | 2.03 | number | -2.90
power | 1.76 | adj | -2.73
negate | 1.74 | prep | -2.14
informal | 1.66 | risk | -1.93
focuspast | 1.62 | article | -1.91
reward | 1.57 | anx | -1.76
achiev | 1.53 | compare | -1.68
pronoun | 1.48 | ingest | -1.52
pro1 | 1.46 | affiliation | -1.48
Table 12: Lexical Analysis on our synthetic data: Log-odds-ratio of LIWC
categories between Asians and African Americans. LIWC categories that also
matches the topics in the UMD-ODH data are highlighted in bold