# A new degree of freedom for opinion dynamics models: the arbitrariness of
scales

Dino Carpentras1,2 Alejandro Dinkelberg1,2 Michael Quayle1,3
1Social Dynamics Lab, Department of Psychology,
Centre for Social Issues Research, University of Limerick, Ireland
2MACSI (Mathematics Applications Consortium for Science and Industry),
Department of Mathematics and Statistics, University of Limerick, Ireland
3Department of Psychology, School of Applied Human Sciences,
University of KwaZulu-Natal, Scottsville, South Africa

###### Abstract

Opinion dynamics models have been developed to study and predict the evolution
of public opinion. Intensive research has been carried out on these models,
especially exploring the different rules and topologies, which can be
considered two degrees of freedom of these models. In this paper we introduce
what can be considered a third degree of freedom.

Since it is not possible to directly access someone’s opinions without
measuring them, we always need to choose a way to transform real world
opinions (e.g. being anti-Trump) into numbers. However, the properties of this
transformation are usually not discussed in opinion dynamics literature. For
example, it would be fundamental to know if this transformation of opinions
into numbers should be unique, or if several are possible; and in the latter
case, how the choice of the scale would affect the model dynamics.

In this article we explore this question by using the knowledge developed in
psychometrics. This field has been studying how to transform psychological
constructs (such as opinions) into numbers for more than 100 years.

We start by introducing this phenomenon by looking at a simple example in
opinion dynamics. Then we provide the necessary mathematical background and
analyze three opinion dynamics models introduced by Hegselmann and Krause.
Finally, we test the models using agent-based simulations both in the case of
perfect scales (infinite precision) and in the case of real world scales.

Both in the theoretical analysis and in the simulations, we show how the
choice of the scale (even in the case of perfect accuracy and precision) can
strongly change the model’s dynamics. Indeed, by choosing a different scale it
is possible to (1) find different numbers of final opinion clusters, (2)
change the mean value of the final opinion distribution up to a change of $\pm
100\%$ and (3) even transform one model into another.

We conclude our analysis by discussing how the scale acts as a new (and still
unexplored) degree of freedom for opinion dynamics models. This result affects
both researchers interested in exploring the properties of different models
and those interested in using them in practical cases. Exploration of its
effects on different models is a key step in understanding the properties of
these models and how they could be successfully employed in real-world
applications. Thus, in our conclusion, we list a series of possible studies
which will expand our knowledge on this degree of freedom and speed up the
process of connecting the models to real-world data.

_K_ eywords opinion dynamics $\cdot$ agent based models $\cdot$ agent based
simulations $\cdot$ empirical validation $\cdot$ degrees of freedom $\cdot$
scale $\cdot$ psychometrics

##  1 Introduction

###  1.1 Opinion dynamics and its degrees of freedom

In recent years, more and more opinion dynamics models have been developed and
studied. These models use a population of interacting entities, usually called
agents. Each agent has at least one variable representing her own opinion
(Castellano et al. 2009; Flache et al. 2017). The opinion has been represented
as a continuous, discrete (ordinal) or nominal variable, both in
monodimensional and multidimensional spaces. The space in which an opinion can
be found (usually a subset of the real numbers) is referred to as the opinion
space. For the rest of this paper, we will focus on one of the most popular
choices, which is the case of continuous monodimensional opinions. For
example, representing an agent’s opinion in a continuous opinion space ranging
between zero and one.

Some attempts have been made to generalize these models, especially to
understand what makes different formalisations similar or events identical
(Coates et al. 2018). At the moment, the two main degrees of freedom are the
interaction rule and the network topology.

In the network, nodes represent the agents while edges represent some kind of
connection between agents. Usually if no edge exists between two agents it is
not possible for those agents to interact, whereas if an edge exists,
interaction between these agents is possible (depending on the interaction
rule). The interaction rule, specifies how agents interact with each other.
That is, how they choose partners for the interaction and how they update
their opinion after the interaction has taken place. In this article, we show
that the decisions made about how to represent psychological constructs as
numbers constitute an additional degree of freedom.

###  1.2 Representing opinions using numbers

Representing opinions as ordered numbers has become common in our society
(Danziger 1994). Psychometric scales are routinely used to convert a construct
(such as someone’s opinion or intelligence) into numbers. Indeed, it is common
to assume a correspondence between psychometric measures and numbers in
opinion dynamics models. The assumption is general enough that usually authors
do not need to discuss the properties of the model’s opinion space (Castellano
et al. 2009; Flache et al. 2017), or can directly use psychological data as
input for their models (Duggins 2017; Jia et al. 2015; Innes 2014).

A fundamental feature of psychometric scales is that the choice of the scale
itself is an arbitrary step, meaning that is possible to find several
different scales which are equally valid for measuring the same construct. A
similar situation can be seen in physics, where usually we do not have any
criterion for claiming that one unit is better than another (e.g. meters vs
feet). However, in physics this process is generally limited to linear
transformation of the same units (e.g. $1m=3.3ft$), while psychological scales
also admit nonlinear transformations. For example, by squaring a unit of
distance, we will obtain a unit of area. However, in psychometrics if $x$ is a
scale measuring opinions on vaccination, $x^{2}$ is considered a scale
measuring the same construct on a transformed axis. Indeed, in empirical
social science research it is recommended practice to re-scale psychometric
data using nonlinear functions (e.g. using $x^{2}$ or exponential
transformation when compensating for skewness)(Tabachnick et al. 2007; Howell
2012).

Another way to investigate this peculiarity of scales, comes from the fact
that in in the physical world we can combine two objects to obtain a new one.
Thus, we can have two rods of the same length and, after combining them, we
obtain a rod twice as long as the original one. Indeed, we can say that a 2
meters object is twice as long as a 1 meter one. This operation is not
possible with opinions; it is not possible to combine two identical opinions
to obtain an opinion twice as strong as the original ones. Consequently, if
person A scored 2 and person B scored 4 on an opinion scale, we cannot say
that B’s opinion is twice as strong as A’s one. However, these scales are
still giving us some kind of information. We can still say that B’s opinion is
stronger than A’s one. All the scales that preserve this order (i.e. A’s
opinion < B’s opinion) are commonly referred to as ordinal scales111Here we
assumed that ordinality is preserved across participants. This is often
referred to as inter-individual measurement invariance. If this does not
exist, then it is possible that measures are still ordinal but within
participants (e.g. the same participant at two time points)..

Notice that is also possible to have scales which are not ordinal. For
example, if we are supposed to distinguish between people who prefer
basketball, baseball or football we would use what is called an nominal scale.
Indeed, it does not make sense to establish a hierarchy between these
categories (e.g. people who prefer basketball > people who prefer baseball).

Since the only requirement for ordinal scales is to preserve order, when these
scales undergo transformations they will still be ordinal scales (as long as
the transformation is an order preserving one). For example, $2<3$ and
$2^{2}<3^{2}$. Thus it is possible to use both linear transformations and non
linear transformations such as $x^{2}$, square root, logarithmic, exponential,
etc 222Note that some transformations, such as $x^{2}$, preserve order only
when the scores are positive.. Indeed, most of them are routinely used for
data transformation before applying linear models (Tabachnick et al. 2007;
Howell 2012).

Another brief example of how we can choose different ordinal scales for the
same opinion comes from the use of Likert-type scales. These scales are
obtained from questionnaires which present a statement and offer a group of
possible response options, out of which one is chosen by the participant (e.g.
Bad, Neutral, Good). Usually, response options are coded as evenly spaced
natural numbers for analysis (i.e. [1, 2, 3] for [Bad, Neutral, Good]). The
anchor points are at the discretion of the researcher and are frequently
symmetric around a presumed midpoint. For example, Wada et al. (Wada et al.
2015) use the response options [Very good, Good, Somewhat good, Somewhat bad,
Bad, and Very bad] coded numerically as [1, 2, 3, 4, 5, 6].

As this is an ordinal scale, it is neither clear that the response options
"good" and "bad" are equidistant from the presumed midpoint nor that the
modifiers "very" and "somewhat" have similar weight in conjunction with "good"
and "bad." This question was explored in a study by YouGov (Smith 2018) where
people were asked to rate a large number of common response options (e.g.
"Good" or "Bad"). Specifically, people rated how much positive each option was
in an interval ranging from 0 (extremely negative) to 10 (extremely positive).

It is then possible to use these values to see how they compare with the
values used in Wada’s Likert scale. The result is shown in figure 1, where we
can clearly see both the ordinality of the two scales (the bigger the $x$
values, the bigger the $y$ values) and the lack of linearity between the
positive and negative portions of the response scale.

Note also that the YouGov study showed asymmetry in the expressions which are
commonly used as semantically symmetric response options in Likert scales,
such as Good (avg. 6.92) vs Bad (2.60) or very bad (1.76) vs very good (7.90).
This asymmetry can also be seen in the graph as the slope obtained from the
first three answers is different to the one from the last three (see the
dashed lines).

A special case of ordinal scales is the category of interval scales which have
the same properties of the units of measurements used in physics. Indeed,
units of measurement from physics are interval scales (within the framework of
measurement theory) 333Here we do not discuss ratio scales, which are a
special case of interval scales. This is done because ratio scales are not
theoretically possible within psychometrics.. Using interval scale it is
possible to say if opinion A is stronger than opinion B (as in ordinal
scales), and also how much stronger it is (e.g. A’s opinion is twice as strong
as B’s one). Indeed, different interval scales can be obtained from each other
via linear transformation (in the case of zero noise) in the same way as how
we convert meters to feet.

Interval scales were first put forward by Stevens (Stevens et al. 1946;
Stevens 1951), and Luce and Tukey show that they are theoretically obtainable
for psychometric measurements (Luce & Tukey 1964). However it is still not
clear whether it is possible to ever obtain this level of measurement
precision, especially in real-world research settings (Borsboom & Scholten
2008; Kyngdon 2008; Keats 1967; Domingue 2014). Certainly, the requirements
for ensuring interval-level psychometric measurement are very stringent indeed
and rarely met in psychological data (Heene 2013; Borsboom & Scholten 2008).
Therefore ordinal scales are the most common level of measurement for opinion
representation.

To sum up this section, the choice of a scale that represents a specific
opinion is arbitrary. That is, if $S$ is a scale that can be used for
measuring opinions, then $S^{2}$ and $e^{S}$ are also equally valid ordinal
scales. In the extremely rare case involving interval scales, we could either
limit our analyses only to linear transformations (to stay in the same
category), or we could apply nonlinear transformations and obtain ordinal
scales. In the next sections we will show how this possibility of choosing
different scales for the same opinion can be represented as a new degree of
freedom in opinion dynamics models.

Figure 1: Likert scale obtained turning the answers [Very good, Good, Somewhat
good, Somewhat bad, Bad, very bad] into the numbers [1, 2, 3, 4, 5, 6] as used
in Wada et al. These values have been plotted against the respective values
from the YouGov study. The graph shows the lack of linearity between the two
scales, as well as the preservation of the order.

###  1.3 An example of opinion prediction

To better understand how the choice of scale may impact predictions of opinion
dynamics models, let us consider first a naive example. Here we consider two
people with two different opinions on the same subject, say one with a pro-
Trump opinion and the other with an anti-Trump opinion. Several models (such
as the Deffuant model (Deffuant et al. 2000) or the Hegselmann Krause model
(Hegselmann et al. 2002)) assume that if two people interact several times,
their opinions will converge to an average value (let us neglect the
confidence interval for this example). This means that they may end up being
either both pro-Trump, both neutral or both anti-Trump, depending on their
starting points.

Thus, we can consider an opinion space in the interval [0,1]. We choose 0.5 to
represent the neutral point separating the anti-Trump opinion (values < 0.5)
from the pro-Trump opinion (values > 0.5). Supposing two individuals have
opinions of 0.3 and 0.9, the model predicts that they will converge to the
opinion represented as 0.6 (obtained as the average between the two), thus, a
pro-Trump opinion.

As previously mentioned, the representation of opinions as numbers is
arbitrary and in general is not symmetric (showed in the example above
regarding the distance of good and bad from the center). So what happens if we
use different numerical values to represent the same opinions? When measuring
the same people on a different scale, the same two opinions may be represented
as 0.4 and 0.6 (still within the interval [0,1]).

Note here that obtaining different numbers representing the same constructs
could occur for many reasons. For example if, in one scale, a transformation
was applied to compensate for skewness. Or perhaps different phrasing is used
for response options; or questions are phrased differently; or one scale uses
an odd number of response options and the other uses an even number
(Taherdoost 2019).

In any case, if in the second scale represented the opinions as 0.4 and 0.6,
the model will predict convergence to neutrality (average = 0.5). Considering
yet another scale, these same opinions can be represented as 0.2 and 0.6 which
predicts a convergence to an anti-Trump opinion, the opposite to what was
predicted when using the first scale.

Of course, this example is arbitrary and superficial, and it should not be
considered as proof. Nonetheless, it allows us to see how the choice of the
scale may affect the dynamics of a model. Furthermore, it pushes us to ask
several related questions: is this phenomenon due to noise? Or is it
negligible when using many more agents? Does anything change if we include the
confidence interval? In the next sections we will address these questions
showing how the choice of the scale naturally appears as a new degree of
freedom in opinion dynamics models.

Specifically, in section 2 we clarify the concept of scale, to eliminate
ambiguity in the following sections. In section 3 we introduce the
mathematical concept of topological conjugacy, which will be used to test if
two models are the same. In section 4 we will analyze mathematically different
models, especially focusing our attention on three models introduced by
Hegselmann and Krause. In this section we will also show that these three
models are the same under scale transformation. In section 5 we will repeat
our analysis of section 4 using agent based simulations, thus validating
previous results and shining more light on their meaning. In section 6 we will
test the same models using real world scales and showing how the choice of
scale may similarly change the model predictions. Finally, in section 7 we
will summarize our findings and suggest possible follow up studies to further
explore the importance of the choice as scale as a degree of freedom in
opinion dynamics models.

##  2 Definitions and distinctions

###  2.1 Construct

In the previous sections we used the term construct to represent abstract
entities such as opinions or intelligence which can be measured using scales.
Constructs are usually considered to be abstract concepts, thus not something
physical within someone’s brain (Cronbach & Meehl 1955). As we are focused on
the field of opinion dynamics, in this text we will use the terms opinions and
constructs interchangeably. However, even in this field, some models use other
parameters besides opinion, such as stubbornness (Friedkin & Johnsen 1997),
quality of information (Brooks & Porter 2020) and many others (Duggins 2017).
Thus, our findings are not limited to opinions, but to every construct
represented as a continuous variable in opinion dynamics models.

It is also important to distinguish between a construct and its representation
into a scale. The two terms are often confused as we tend to think of a
construct by its numerical value. In the physical world, this would be the
difference between the length of an object (which is a property of the object
itself) and the representation of that length into a scale (e.g. 1 m or 3.3
ft). The fundamental distinction is that changing a scale does not alter the
properties of the object itself (i.e. the object does not become longer or
shorter when we change units). In the same way, when discussing the following
sections, we should keep in mind the difference between someone’s opinion and
its representation on a specific scale.

Since it is not possible to directly access someone’s opinions without
measuring them, we always need to choose a scale. And this is why we are
focusing on how the choice of the scale may affect the model dynamics.

###  2.2 Scales

To avoid confusion in our upcoming analysis, it is necessary to explain what
we mean by the word scale. Researchers interested in the mathematical
properties of scales, usually define them as a relationship between a
construct and a subset of real numbers (Eysenck 1982). Another (and perhaps
more common) definition, relates to the process of building a scale. This
definition states that a scale represents a set of items that can enable the
quantification of a psychological construct. This last definition is the one
that people have in mind when they refer to, for example, the Wechsler Adult
Intelligence Scale (also known as WAIS). Besides the difference of focal
point, the two definitions differ as the second may include finite precision
and accuracy, while the first assumes perfect measurements.

An analysis of how measurement error impacts opinion dynamics models is beyond
the scope of this article. Instead, we are interested in how choosing a
relationship between the construct and numbers can strongly affect opinion
dynamics models. Because of that, we will use the first definition, sometimes
stressing that the analysis refers to the case of perfect accuracy and
precision.

As we mentioned earlier, a scale with perfect precision maps each level of a
construct to a single number. This means that level $l$ will be mapped to
number $n_{1}$ in scale $S_{1}$ and to number $n_{2}$ in scale $S_{2}$.
Consequently, a relationship exists between the two numbers, $n_{1}$ and
$n_{2}$. In the following, we will call $h$ the relationship between the
elements of two scales (see figure 2 for a graphical representation) and we
will write: $n_{2}=h(n_{1})$ (i.e. $h$ transforms $n_{1}$ into $n_{2}$).

In statistical terms, this can be rephrased as the fact that Spearman’s rho
correlation between the two scales is 1 (note that we are not using Pearson’s
correlation as it would assume a linear relationship between scales). In terms
of functions, we can say that a strictly monotonic function can be used to map
one scale onto the other (Krantz et al. 2006a). This allows us to (1) discuss
scales without referring them to a specific construct and (2) define a new
scale $S_{2}$ just by using $h$, its relationship to scale $S_{1}$ (Krantz et
al. 2006a, b, c).

Please, note also that a perfect scale should also have an infinite set of
scores (or at least as many as the participants). This can be counter-
intuitive for people involved in the practical use ordinal scales, as they
usually have a really limited number of possible scores (e.g. a measurement
scale comprising four items with five response options each only has 20
possible scale points). To understand why having a finite number of scores is
not possible in perfect scales, just consider the case of rating 11
participants on a 10 point scale. This implies that at least 2 of them will be
represented by the same score. If the strength of their opinion is not exactly
the same, then the scoring will not be exact and the scale will not be
perfect. This is why perfect scales should have an infinite number of possible
scores within the defined extremes of the scale.

Sometimes in the text, we will also say that a model was seeded with a scale
or that we use a model with a scale (e.g. "we use model $M_{A}$ with scale
$S_{1}$"). This means that the construct was turned into numbers using the
mentioned scale and that those numbers were used as input for the model.

Figure 2: (left) Schematic representation of the relationship between the
levels of a construct (represented as apple size) and the numbers in two
different scales. (right) Relationship between the two scales, represented as
$h$

##  3 Theoretical background

###  3.1 Hypothesis on the opinion space

In order to explore the concept of opinion space scientifically we need to
introduce a testable hypothesis. As mentioned in the introduction, there have
been no previous studies, to our knowledge, that analyze in depth the concept
of opinion space, so we did not find any explicit claims about its properties.
Because of that, here we propose two hypotheses which will help us eventually
reveal hidden assumptions and, in general, better understand opinion dynamics
models.

Hypothesis 1 (H1) is based on the idea that the choice of the scale should not
affect the model dynamics. Formally, we write it as: All psychological scales
measuring the same construct with infinite accuracy and precision produce
equally valid results when seeded to the same opinion dynamics model.

The practical meaning of H1 is that, whatever the scale, the model will
produce the same result (we will properly define the concept of same result in
the next section). Or we can say that, as long as the same construct is being
(perfectly) measured from the same people, every scale can be used
equivalently as model’s input. If H1 turns out to be false, it means that,
even if the two scales are perfect, they will produce different results when
used in the same model.

The second hypothesis is based on the idea that it should be possible to find
a scale (such as interval scales) which should work with every model. This
would need to be true in order to directly compare different models without
discussing the properties of their opinion spaces, as is common practice in
review articles (Castellano et al. 2009; Flache et al. 2017). Phrased
differently, this hypothesis tests the idea that one opinion space can be used
equivalently across multiple models. Thus, the first version of this
hypothesis (H2.1) states that: “given two models $M_{1}$ and $M_{2}$, whose
opinion spaces are defined on the same interval, and capable of producing
results with the same precision, if scale $S$ works perfectly for the model
$M_{1}$ then it also works perfectly for model $M_{2}$.”

We can also introduce a more general version of this hypothesis (H2.2)
claiming that: “if scale $S$ works perfectly for model $M_{1}$ then it also
works perfectly for model $M_{2}$, if the opinion spaces of the two models are
defined on the same interval.”

In the following, we will analyze both versions of H2, making clearer the
difference between the two. For now, let us just notice that, if H2.1 and H2.2
turn out to be false, we could find a scale $S_{1}$ which is the best scale
for model $M_{1}$, and a different scale $S_{2}$ which is the best scale for
$M_{2}$ (even if they are perfectly measuring the same construct). In terms of
opinion space, this would mean that the opinion spaces of the two models are
different (meaning that they require different scales).

###  3.2 Topological equivalence

When discussing H1, we mentioned the idea of finding the same results while
using two different scales on the same model but did not explain what we meant
by the same result. A simple (but wrong) way to express this concept, would be
to expect the same numeric output in the two cases. To understand why this is
a poor approach, just consider the case of a toy model which outputs the mean
height of two objects. If we input $1m$ and $2m$ we obtain an average of
$1.5m$. While if we input the height of the same objects in feet, we obtain an
average of $4.9ft$. This last value, is equivalent to the $1.5m$ we just found
but not numerically identical.

What we have just shown is that, when changing scale, we do not expect to
obtain the same numbers as output (indeed, $1.5\neq 4.9$). However, we expect
to have a way to convert both the input and the output from one scale to the
other.

In the same way, with opinion dynamics models we do not expect to obtain the
same numeric output when working with two different scales. Instead, we expect
to be able to convert the outputs from the model when using different scales
(see figure 3). Thus, making the choice of the scale independent from the
meaning of the result. On a qualitative level, this can be seen as the fact
that if the model predicts people converging to pro-Trump opinions when using
scale $S_{1}$, we expect to make the same prediction with scale $S_{2}$ as
well, even if this result is represented with different numbers.

To mathematically express this concept (and so make it scientifically
testable) we use the concept of topological equivalence (Alligood et al.
1998). To avoid overloading this section, we suppose that the model is
deterministic (thus without random variations). We also suppose that the model
produces a single output, represented as $M(x)$, where $x$ is the input data.
Since these requirements are extremely stringent, we will relax them in the
next section, thus extending our analysis to all continuous opinion dynamics
models.

To be precise, we should also distinguish between the concept of input data
and the scale used to produce those data. For example, we could find that two
people have an opinion respectively of $0.2$ and $0.6$ on a scale. In this
case, the two numbers $0.2$ and $0.6$ would be the data, while the
relationship between the construct and those numbers would be the scale. Since
this paper assumes perfect measurement with no noise or measurement artifacts,
this distinction will not improve clarity in the present analysis. Thus, for
simplicity, we will use $S_{i}$ to mean both the scale and the data measured
in scale $i$.

With these premises, we can write the topological equivalence between two
models as the equivalence of the model’s output with the two scales as:

$M(\mathbf{S_{2}})=h(M(\mathbf{S_{1}}))$ (1)

Where bold is used to represent vectors, $S_{1}$ is the data in scale $1$ and
$S_{2}$ the same data measured in scale $2$, $M$ the function which produces
the model’s output and $h$ is the (already mentioned) transformation from one
scale to the other, such that:

$\mathbf{S_{2}}=h(\mathbf{S_{1}})$ (2)

Notice that the left term of 1 is the result of using the model with scale $2$
(in figure 3, this would be moving from top to bottom on the right side).
While the right term is obtained by using the model with scale $1$ and then
converting the result using $h$ (thus moving from top to bottom on the left of
figure 3 and then moving to the right). Finally, 2 represents how $S_{2}$ can
be obtained from $S_{1}$ (from top left to top right in figure 3).

Note also that we are using $h$ acting both on scalars and vectors. This is
because for every $\mathbf{x}$ $\epsilon$ $R^{N}$, $h(x)$ is found as:

$h([x_{1},x_{2},…,x_{N}])=[h(x_{1}),h(x_{2}),…,h(x_{N})]$ (3)

If equation 1 holds, we say that using scale $1$ is equivalent to using scale
$2$ with model $M$. This means that we can convert both the model’s input and
output from one scale to the other, obtaining the same results (as we did with
meters and feet in our toy model).

In the next sections, we will compare also the case of running two different
models, $M_{1}$ and $M_{2}$, on two different scales. In this case 1 can be
generalized to:

$M_{2}(\mathbf{S_{2}})=h(M_{1}(\mathbf{S_{1}}))$ (4)

If equation 4 holds, we say that using scale $1$ with model $M_{1}$ is
equivalent to using scale $2$ with model $M_{2}$.

To see how 1 works with a model, let us recall the example of the model of the
mean height (represented in figure 3). In this case we are using the same
model with two different scales. Thus, we use equation 1 with $M(x)=mean(x)$.
Furthermore, we have $h(x)=3.3x$, which is the conversion between the two
scales ($1m=3.3ft$). If we substitute all of this into equation 1 we find the
same values on the left and on the right hand side, meaning that using meters
or feet is equivalent for this model. Also, this can be stated as the fact
that, it does not matter if we make calculations in meters or feet, as we can
always convert one into the other using $h$.

Coming back to the example of Trump-related opinions we suppose that the two
opinions are $[0.4,0.6]$ which produces an average of $0.5$. If we use
$h(x)=10x$, we will have opinions $[4,6]$, which produce an average of $5$.
Since $h(0.5)=5$, we see that we convert the two results into each other using
$h$. Thus, we can use any of the two scales without affecting the result.

Figure 3: A scheme of how $h$ converts $S_{1}$ to $S_{2}$ both at the input
and at the output of the two model instances. The input in meters (top left)
can be converted into feet (top right) using the conversion function (light
blue). The model will output two different numbers depending on what we
provided as input (either meters, left or feet, right). However, by using the
same conversion function, we can transform one output into the other. Thus, it
does not make any difference if we use the model with one scale or the other.

##  4 Theoretical analysis

Here we analyse under which cases equation 1 holds for opinion dynamics
models. Thus, in which cases the scale transformation do not affect the
model’s output. To make the analysis more sound we relax the hypothesis of the
model being deterministic and producing a single number as output. This can be
done by requiring $M$ to represent a single step of the model, instead of
being the final output. This means that $M(x)$ is just the updated opinion
value of one agent. For example, if at each step an agent updates her opinion
with the average of all the other agents’ opinions, we will have
$M(\textbf{x})=mean(\textbf{x})$, where x denotes the vector containing the
opinion of the agents selected for the interaction.

We can easily see that equation 1 does not hold for every possible function.
We can verify this, for example, by choosing $M(x)=2x$ and $h(x)=x^{2}$.
However, by assuming specific functions it is also possible to find conditions
under which the equation holds. For example, by assuming both $h$ and $M$ as
linear functions, the equation is satisfied. Another interesting case in which
equation 1 is satisfied is when an agent copies the opinion of another agent
(such as in the voter model or in the Axelrod model). In this case equation 1
is valid for both linear and nonlinear transformations.

In general, we could say that non-arithmetic models (i.e. models which rely
uniquely on copying) are not altered by scale transformations. Instead,
arithmetic models (i.e. models which are based on averaging or other
arithmetic processes) may produce different results when their scale is
transformed.

This means that if we want to know all the arithmetic models that satisfy
equation 1 (and so are unaffected by scale transformations), we should analyze
them individually. Since a review of every model is beyond the scope of this
paper, here we focus on just three models, which have been presented by
Hegselmann and Krause (Hegselmann & Krause 2005). This will be enough to test
the hypotheses and, at the same time, provide some insight into the importance
of scale selection in opinion dynamics modelling.

###  4.1 Definition of the models

In the models we are going to analyze every agent has an opinion $x$ defined
by a scalar variable in the interval $[0,1]$. Furthermore, the model is
characterized by a confidence interval $\varepsilon$. At each step:

  1. 1.

An agent $i$ is selected

  2. 2.

From all her neighbors, only those within her confidence interval
$\varepsilon$ are selected

  3. 3.

The agent updates her own opinion averaging the opinion of all her selected
neighbors (including herself)

In the original paper, these models have been studied on complete graphs in
the case of synchronous update. In our case we keep the network topology, but
we study the models in cases of both synchronous and asynchronous update.
Indeed, synchronous modelling has been shown to produce artifacts in some
models (Galán et al. 2009), and we want to make sure that our results are not
due to artifacts.

The three models differ from each other by their averaging function, leading
to the following three functions (one for each model):

$M_{A}(x)=\frac{1}{n}\sum_{i}x_{i}$ (5) $M_{G}(x)=\sqrt[n]{\prod x_{i}}$ (6)
$M_{H}(x)=\frac{n}{\sum_{i}\frac{1}{x_{i}}}$ (7)

As these functions are measurements of mean, their output number is always
between the minimum and the maximum of the input data.

To simplify the theoretical analysis, in this section we analyze the case in
which the confidence interval is the maximum for all three models. Meaning
that everyone can interact with everyone else. This simplification will be
removed in section 5, where we will study these models via simulations using
the complete model specified in the original paper.

As mentioned, the opinion space is defined as the interval between $0$ and
$1$. However, below we will also study what happens when we expand this
interval (e.g. between $1$ and $100$). This is another procedure that has been
previously performed in literature (Gargiulo & Gandica 2016) that is worth
considering. Thus, here, we will use topological equivalence to explore why
this procedure might produce different results.

###  4.2 Analysis of H1

H1 was based on the idea that using different scales on the same model will
provide the same result. As previously discussed, when $M$ and $h$ are linear,
this is true (equation 1 holds), while this may not be the case with nonlinear
models. For example, let us define a scale $S_{1}$ defined over the interval
$[0,1]$. Then, let us introduce $S_{2}$ defined by $h_{1\rightarrow
2}(x)=99x+1$, which maps the interval $[0,1]$ of $S_{1}$ to the interval
$[1,100]$ of $S_{2}$. In this case equation 1 holds perfectly for $M_{A}$,
while it does not for $M_{G}$. To quickly confirm this last point, just
consider two test values on $S_{1}$: $0$ and $1$. By substituting them in
equation 1, for $M_{A}$ we obtain the same result on the left and on the right
hand side. This is not true for $M_{G}$, where we find $1$ on the left hand
side and $100$ on the right.

This means that, while data on linear models can be linearly rescaled (and so
the interval can be linearly extended), this is not true for nonlinear models.
Indeed, these models can produce different results when using linear
transformations of the same scale as input. Just to stress this concept,
imagine the case of some opinion data in a scale $1$ to $7$ and the problem of
how to fit them into $M_{G}$. A possibility would be to linearly rescale the
data between $0$ and $1$, to preserve the model’s interval. Alternatively, we
can preserve the data values and just run the model in the interval $1$ to
$7$. While the two procedures would be equivalent on a linear model like
$M_{A}$, they will produce different results on nonlinear models such as
$M_{G}$.

However, it is important to notice that this does not mean that rescaling is
problematic only for nonlinear models. Indeed, by using $S_{3}$, defined by
the function $h_{1\rightarrow 3}(x)=x^{2}$ even $M_{A}$ does not produce the
same results. We can show this using two test numbers such as $1/4$ and $3/4$.
This means that if we plan to use $M_{A}$ we can linearly rescale data, while
any other transformation (such as compensation for skewness) will alter the
model dynamics.

Another important comment regards the choice of the scale. We know that
ordinal scales can be thought as nonlinear transformation of each other (as
shown in figure 1). Thus, using different scales will produce different
results even on linear models. To go back to the example for Wada’s data, we
could choose whether we want to represent them using the classical numbering
of Likert scales (1 to 6) or the ratings of positivity from the YouGov study.
As we will show in section 6, this choice would produce different results when
we run the model and there is no easy way to say which choice is more
appropriate. The only way would be to test them and compare the their
predictions with the dynamics of the real world.

###  4.3 Analysis of M2

For the following analysis, we consider the ideal case in which we found a
scale $S_{1}$ which works perfectly with $M_{A}$ (i.e. makes predictions which
perfectly adhere to reality) and we wonder if the same scale can be used for
all the other models. This would mean that the scale that works best for one
model, also works best for all the other models.

Just by knowing that $M_{A}(x)\neq M_{G}(x)$ we can already tell that the two
models will produce different results when using the same scale, thus H2.2 is
false. However, this may just mean that $M_{A}$ is correct, while $M_{G}$
makes poor predictions. To prove this false, in the next lines we will show
that if $M_{A}$ works perfectly, then even $M_{G}$ works perfectly, but on a
different scale.

Let us consider scale $S_{4}$ defined by the transformation $h_{1\rightarrow
4}(x)=e^{x}$. Since the exponential function transforms the sum into a
product, it is possible to show that:

$M_{G}(h_{1\rightarrow 4}(S_{1}))=h_{1\rightarrow 4}(M_{A}(S_{1}))$ (8)

Which can be rewritten as:

$M_{G}(e^{S})=e^{(M_{A}(S))}$ (9)

This equivalence means that if we use $M_{G}$ with $S_{4}$ we will always
obtain the same results as using $M_{A}$ with $S_{1}$. Furthermore, the same
results can be obtained with $M_{H}$ using $h_{1\rightarrow 5}(x)=1/x$ (i.e.
using $M_{H}$ with $S_{5}$ is equivalent to using $M_{A}$ with $S_{1}$).

This shows two things. The first is just that H2.1 is false: we cannot find a
single scale which is the best for every model. The second important
consequence of this results is that we can find three different models
($M_{A}$, $M_{G}$ and $M_{H}$) each one producing perfect results within its
own scale ($S_{1}$, $S_{4}$ and $S_{5}$ respectively). This also means that
$M_{A}$, $M_{G}$ and $M_{H}$ are the same model under a scale transformation.

###  4.4 Model invariance

As discussed before, linear models can be linearly rescaled, while nonlinear
models, in general, cannot. We can see this applying a linear transformation
on $M_{A}$, such as:

$M_{A}(\alpha x+q)=\alpha M_{A}(x)+q$ (10)

Which is just the application of equation 1 with $h(x)=\alpha x+q$ and allowed
us to say that the extension of the opinion space in linear models (as done in
(Gargiulo & Gandica 2016)) is legitimate. While this linear rescaling does not
work on nonlinear models, such as $M_{G}$, it does not mean that their
interval cannot be extended or rescaled. Indeed, notice that

$M_{G}(\beta x^{n})=\beta[M_{G}(x)]^{n}$ (11)

This means that $M_{G}$ is invariant to rescaling of the type $\beta x^{n}$.
This appears more natural if we remember that $M_{G}$ is the exponential
transformation of $M_{A}$. Thus, the linear rescaling of one model is
transformed in the exponential rescaling of the other. This also solves the
problem of how we can rescale the data for nonlinear models.

As we will discuss in the final section, this property may be useful not just
to know how to rescale each model, but also for finding equivalence between
different models.

##  5 Analysis via agent-based simulations

###  5.1 Method

In the previous sections we performed a theoretical analysis of the
aforementioned Hegselmann Krause models. However, to simplify the analysis, we
added some important constraints. The first one was performing the analysis
using the maximum possible confidence interval. Even if this case is allowed
by the original models, it is also a special case. Indeed, by allowing every
agent to interact, we obtain a model with no real confidence interval.

The second major assumption stated that two models are considered equivalent
if they produce the same result at every step. This could be considered a
stringent requirement. Indeed, we may be interested just in the case where two
models always produce the same output result, without any restriction on the
dynamics.

Furthermore, in our previous analysis we often used just two numbers to test
our hypotheses. However, in standard simulations we have (1) random
fluctuations and (2) many more agents. So, it is possible that, when using a
large number of agents, the effect of using different scales is completely
hidden by the random fluctuations of the model itself. Another way to state
this would be to notice that, even if two models are not mathematically
identical, their difference during use may be so small as to be negligible.

To analyze this possibility and weaken our previous assumptions, we perform
simulations on two of the previously mentioned models (specifically $M_{A}$
and $M_{G}$). To make sure that our results are not due to a small number of
agents, we performed simulations with $N=10,000$. This is way higher than most
publications on agent-based models. Following the standard introduced by the
authors of the model, we used a fully connected graph as topology and all
agents were initialized with a random opinion from a uniform distribution.
Hegselmann and Krause tested their models using synchronous updating, but, as
this technique has been shown to produce artifacts in some models (Galán et
al. 2009), we tested the case of both synchronous and asynchronous updating.
The synchronous case was run with a maximum of $100$ iterations, each
iteration updating all the $10,000$ agents. The asynchronous case was run
$100*10,000$ iterations, each iteration updating only one random agent. As
observed also by Hegselmann and Krause, this was more than enough for the
model to converge. In order to collect enough random variations, we repeated
each run $100$ times, each time collecting the average of the final
distribution. As the average changed slightly each time (due to random
variations), this resulted in a distribution of averages (see figure 4).

For standard $M_{A}$ we used an opinion space between $0$ and $1$, and a
symmetric confidence interval defined by $\varepsilon=0.2$. When we used scale
transformation, we transformed both the space and the confidence interval,
obtaining an asymmetric interval. This has been done as the confidence
interval determines the agents that are selected for the interaction. By using
this transformation, there is no difference in the selected agents when using
each one of the scales. Indeed, both the opinion space and the confidence
interval should be measured with the same scale, thus it would not make sense
to transform only one of them.

Agents’ opinions were initialized with random numbers from a uniform
distribution. Scale transformations were applied after initialization.

###  5.2 Results

For each iteration, we collected the average of the final opinion
distribution. In figure 4 we plotted the distribution of these averages when
running the model $100$ times using the same parameters. Curves have been
smoothed using the kernel density estimate provided by the Seaborn package for
Python (Waskom 2020). Results were almost identical for the case of
synchronous and asynchronous update, thus we decided to plot only the
asynchronous case. To compare the results, we transformed each distribution
using the inverse function (in agreement with the concept of topological
equivalence, equation 1). This allowed us to have all the final distributions
in the $[0,1]$ interval. Without this transformation each scale would be in a
different interval (e.g. the logarithmic one would be in the interval $[1,e]$)
making impossible to compare the final results. This process is further
detailed in the next section.

To confirm the theoretical analysis and the idea of an asymmetric threshold,
we run both $M_{A}$ with no scale transformation and $M_{G}$ with the
exponential scale transformation. According to the theoretical analysis, these
two models should be equivalent. As shown in figure 4, the two curves overlap
almost completely, in agreement with our expectations.

To observe the effects of different scales on $M_{A}$, we used five different
transformations: linear, $x^{2}$, square root, exponential, and logarithmic.
Curves show almost no overlap (thus producing different predictions) except in
the case of the linear rescaling, which, in agreement with the theory, did not
change the model’s output. We run a similar analysis also for $M_{G}$ showing
that is indeed affected by linear rescaling but not from transformation of the
kind $\alpha x^{n}$. Also this is in good agreement with our theory.

By looking at the graph of $M_{A}$, we could be tempted to state that the
uncertainty on the scale leads to an error of roughly $\pm 5\%$. However, this
result may change by using different transformations. For example, by looking
at scale transformations of the type $x^{n}$ we can find the following limits
for $M_{A}$:

$\lim_{n\rightarrow\infty}\left[mean(\textbf{x}^{n})\right]^{\frac{1}{n}}=\max{[\textbf{x}]}$
(12)

$\lim_{n\rightarrow\infty}\left[mean(\textbf{x}^{-M})\right]^{\frac{-1}{n}}=\min{[\textbf{x}]}$
(13)

The first equation means that as we use transformations with higher n (in the
scale transformation $x^{n}$) the output of $M_{A}$ moves gradually from 0.5
to 1. We tested this assumption running $M_{A}$ with the transformation
$x^{10}$ and obtaining an average of $0.76$ (thus an increase of roughly
$50\%$ to the case of $M_{A}$ with no transformation). This is also shown if
figure 4. A similar result can be obtained using the transformation $x^{-n}$
which, according to equation 13, will move the final average of the model from
$0.5$ to $0$. This means that if we allow all possible scale transformations,
our uncertainty would be of $\pm 100\%$ as the convergence number could be
anywhere in the entire interval $[0,1]$.

As we will outline better in the next section, it is true that we do not know
which is the best scale for representing the data (in terms of opinion
dynamics models) but it is also true that we will not expect to consider
transformations such as $x^{100}$. Thus, we should be able to find some range
of admissible transformations, which will also result in a limited range of
uncertainty on our models. Identification of similar ranges is an entire new
study and beyond the scope of this paper. Indeed, we expect it to depend on
the model itself, as well as on the network topology which can amplify or
reduce this uncertainty.

Figure 4: Distributions of the average opinion. Each configuration
(represented by a single curve) was repeated 100 times. Curves have been
smoothed using a kernel density estimate. $M_{A}$ predictions are the same
under linear rescaling, while they change for nonlinear transformations.
$M_{G}$ instead changes when linearly rescaled, but is invariant to
transformations of the type $x^{n}$. $M_{A}$ without scale transformation is
equivalent to $M_{G}$ with the scale transformation $h=e^{x}$

###  5.3 Further clarification

Some readers may think that the difference in the qualitative result is not
surprising. Indeed, coming back to our Trump-related example, we may imagine
to find an output average of $0.6$, thus predicting a pro-Trump opinion. Now,
if we consider the scale transformation $x^{2}$, we may think that the model
should converge to $0.6^{2}=0.36$. As this is below $0.5$ this should predict
an anti-Trump average opinion.

As mentioned in section 3, this line of thinking is wrong, or, at least, it is
not what we studied in this paper. Indeed, when we transform from one scale to
the other, we transform all the points (including the threshold separating
pro-Trump from anti-Trump opinions). So, in this case, everything below
$0.5^{2}=0.25$ will predict a pro-Trump opinion.

To show this, let us imagine that the model ran with the second scale
converged to 0.16. Since 0.16<0.25 we could already tell that here the model
is converging to an anti-Trump opinion. However, to better compare this result
with the first, we have to transform it back to the original scale. Since we
squared the scale, to transform it back we have to use the inverse function,
obtaining $\sqrt{0.16}=0.4$. Thus this (hypothetical) model will predict a
result of 0.6 with the first scale and 0.4 with the second scale.

Figure 5: Representation of how the the threshold is transformed together with
the opinion transformations and how different simulations are compared.

##  6 Using real world scales

So far we have analyzed the case of perfect scales. However, the choice of a
scale affects not only the analysis of the model’s properties, but also their
applications to real data. To show this, here we analyze what happens when we
run $M_{A}$ with two different scales which represent the same construct.

To start, imagine that we have collected some opinion data using the type of
answers shown in Wada’s study: [Very good, Good, Somewhat good, Somewhat Bad,
Bad, Very bad]. Suppose also, that we found the following distribution [2%,
21%, 11%, 29%, 6%, 31%], which means that 2% of people answered "Very good",
21% answered "Good", etc. If we want to run an opinion dynamics model seeded
with these data, we need to choose how to transform the answers to numbers
(i.e. which scale to use). As noted before, we could either use standard
Likert scale coding or an alternative one, such as the one obtained from the
results of the YouGov study (Smith 2018).

If we suppose the maximal confidence interval ($\varepsilon=1$) and we run
$M_{A}$ on the Likert scale we obtain that everyone will converge to opinion:
4.09. Since this is not an interval scale, we cannot claim that our result is
0.09 above "Somewhat bad." Indeed, in ordinal scales we cannot say how much a
value is bigger, but only if it bigger or smaller444Note that we could perform
a similar operation in the previous sections (even if we were dealing with
ordinal scales) as we were working with perfect scales and thus with infinite
points. This means that every output number from the model was still one of
the (infinite) numbers of the scale. This is not true with finite scales,
where only a few numbers belong to the scale.. However, we can still claim
that this result lies between "Somewhat bad" (coded as 4) and "Bad" (coded as
5).

If now we run the same model using response option coding from the YouGov
scale, we find that everyone ends up with an opinion of 5.48. In the YouGov
scale, this value is between "Somewhat good" (3.16) and "Somewhat bad" (5.82),
which is different from the previous interval.

This shows how the same construct from the same people may lead to different
predictions when using two different scales (in this case, even using the same
questionnaire). However, we still do not know how rare this situation is. In
the next section, we show the probability of making different predictions
while using these two different scales.

###  6.1 The practical effect of real world scales

In order to measure how rare is the case described in the previous section, we
simulated different distributions. Each distribution was obtained using six
random variables, each one representing the amount of people who selected one
of the six answers (i.e. the first variable represented the fraction of people
answering "Very good"). In this way we can simulate every possible
distribution.

After, we initialized 1,000 agents accordingly to the previously obtained
distribution. Two simulations were then run, one using conventional Likert
scale and another using the scale from the YouGov study. Both scales were
linearly renormalized between 0 and 1 before the simulation started. The two
simulations used the same network (fully connected) and same confidence
interval.

For each run we collected the interval of the final average (e.g. between
"Good" and "Somewhat good") and the the number of clusters (e.g. 1 if everyone
shared the same final opinion, 2 if two final opinions were obtained, etc).
Finally, we compared how often using the two scales brought to the same
results.

Table 1 shows that the two scales predict the same interval for the final
average roughly 78% of the times, almost independently from the confidence
interval. The situation is more complex when we consider the number of
clusters. Indeed, depending on $\varepsilon$ we could range from always
obtaining the same result, to never.

| Same interval | Same num. of clusters
---|---|---
$\varepsilon=1$ | 78% | 100%
$\varepsilon=0.4$ | 77% | 97%
$\varepsilon=0.3$ | 78% | 46%
$\varepsilon=0.2$ | 79% | 61%
$\varepsilon=0.15$ | 78% | 0%
Table 1: Percentage of times the model produced the same results with the two
scales

This section shows practically that the use of different scales may have a
serious impact not only on the properties of the model, but also on the
meaning of its predictions. Notice also that here only two scales were
studied, while almost infinite possibilities would be available. A full study
of how different scales would affect $M_{A}$, or any other model, is out of
the scope of this paper. Here we only wanted to show that this effect exists
and that it is not negligible.

##  7 Discussion and future studies

Opinion dynamics models have the potential to be applied to everyday life, as
they could allow to simulate the effect of different policies, like a digital
twin of our society (El Saddik 2018). To reach such a result, better knowledge
is required about how the models work and how they should be used with real
world data. This paper addresses both of these two aspects.

Indeed, up to now modelers studied the effect of the network topology and the
interactions rules to understand the model’s dynamics. Here we showed that a
third degree of freedom is also present: the scale. Here we showed how
changing the scale we can (1) strongly change the model’s dynamics, up to
completely changing the result (equivalent to an uncertainty of $\pm 100\%$),
(2) turn one model into another (both of them in the standard literature) and
(3) find how to rescale data for nonlinear models. This shines new light on
the problems of (a) exploring models’ properties and (b) validating models
against real data.

People interested in model validation should notice that there is no reason to
believe that the scales most commonly used in psychometrics will provide the
best predictions for a specific model. Therefore, these scales, when used in
opinion dynamics models, could be considered transformations of an unknown
best scale which should be found for each model. This problem could be solved
in many ways.

A first method would be to just fix a scale and test the models against it.
This would change the problem of "finding the best scale given a model" to
"finding the best model given a scale." Another method would consist of
looking for explicit or implicit model specifications. For example, the
Deffuant model (Deffuant et al. 2000) is such that people will always move to
an average position. Meaning that if a person with opinion $0.5$ interacts
with a person of opinion $1$, both of them will move the same amount (e.g.
$0.1$) towards the average (i.e. $0.75$). Knowing this, it is possible (in
theory) to design a Deffuant scale in which people, after interaction, move by
the same amount. Furthermore, new models could be designed with stronger
assumptions about the nature of the scale or the opinion distribution. This
would help researchers to identify the right scale, or directly control this
degree of freedom.

Alternatively, some researchers may prefer non-arithmetic models, (i.e. models
which rely on copying other agent’s opinion). Indeed, we showed how this rule
satisfies equation 1 and thus is insensitive to scale transformation. One
example is Axelrod’s model of cultural diffusion (Axelrod 1997), which can
also be extended to resemble bounded confidence models, using ordinal scales
instead of nominal (MacCarron et al. 2020).

People interested in the study of model properties could characterize the
models’ invariances. Thus, which models can be linearly rescaled and which,
instead, should use different rescaling, such as exponential rescaling.
Eventually, it may be possible to find models which cannot be rescaled, which
will pose a new challenge on how to process data. Similarly, models like
$M_{G}$, can be only exponentially rescaled, meaning that the entire set of
data should be either positive or negative. This poses a problem regarding
what to do with scales which originally include both positive and negative
values (such as a range $[-10,10]$).

Research should also focus also on how each model is affected differently by
the scale uncertainty. As mentioned in section 5, allowing all possible scale
transformations may produce an uncertainty of $\pm 100\%$. However, we do not
expect to be so uncertain about the nature of the scale to consider
transformations such as $x^{100}$, thus we should identify which scales are
acceptable. A good starting point for understanding the range of uncertainty
may be the data transformations which are routinely used with psychometrics
data (e.g. linear, square root and logarithmic). Thus modelers may test these
scale transformations and observe how much their model is sensitive to scale
transformation. In the meanwhile, we should also identify the set of
acceptable transformations, to better characterize this process.

Another possible focus would be on ordinal scales, such as Likert scales.
These are the most common type of scales for measuring opinion and have been
already been used with opinion dynamics models (Duggins 2017). However, Likert
scales are nonlinear representations of interval scales (Krantz et al. 2006a),
thus it will be interesting to study the typical "amount of nonlinearity” and
how this may impact the overall result (notice that for carrying out this
analysis we should also define a measurement of nonlinearity). We already
performed a quick analysis in section 6 for $M_{A}$, but different models and
scales should also be tested. Especially, it would be interesting to know if
it is better to use scales with a fewer or more response options in opinion
dynamics models. A similar problem was already studied for psychometric
purposes (Taherdoost 2019), but the best conditions for agent based models may
be extremely different.

Another case of nonlinearity lies in the relationship between models. Indeed,
we observed that $M_{A}$ and $M_{G}$ are the same under a scale
transformation. This also opens the door for the study of models’ invariances.
Indeed, while $M_{A}$ is invariant to linear rescaling, $M_{G}$ is invariant
to exponential rescaling. These properties could be studied to better
understand the model properties, equivalence to other models and the type of
nonlinearity.

Another important aspect to study regards the finite precision and accuracy of
the scales (e.g. noise). This problem has already been introduced by Edmonds
(Edmonds 2004), and further explored by Polhill (Polhill et al. 2005), who
showed possible artifacts (such as the appearance of "ghost agents") when
using floating point numbers. However, to now very little is known about the
interaction between finite precision, network topology and update rule.
Indeed, they could possibly interact with each other, amplifying or limiting
the overall uncertainty.

Similarly to the effect of finite precision, data can also include biases
which can affect the model. This has been observed in deep learning, where
just the sampling of data is enough to introduce race-based or gender-based
biases (Kay et al. 2015). Thus, when considering applications of opinion
dynamics models to the real world, this possibility should also be considered.

Finally, data analysis may also have a key role in the evolution of opinion
dynamics models. Indeed, the field of psychometrics up to now has generally
developed scales aimed at psychological research or clinical applications. The
current paper shows a gap between psychometric best practice and agent based
models, which eventually could be bridged by producing data which optimized
for opinion dynamics models.

Here we showed how the choice of a scale for representing data can strongly
affect the opinion dynamics. This effect is strong enough to completely change
the model’s output or transform one model into another. Thus, similarly to the
update rule and the network topology, we think that the scale can be
considered as a new degree of freedom for opinion dynamics models.

## Acknowledgements

We would like to thank James Gleeson, Padraig MacCarron, Paul J. Maher,
Caoimhe O’Reilly, Kevin Burke and Susan Fennell from University of Limerick,
Ireland for the fruitful discussions.

This preprint is part of a project which has received funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No 891347.

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