# Modelling opinion misperception and the emergence of silence in online
social system

Daniele Vilone1,2, Eugenia Polizzi1

1 LABSS (Laboratory of Agent Based Social Simulation),
Institute of Cognitive Science and Technology, National Research Council
(CNR), Rome, Italy
2 Grupo Interdisciplinar de Sistemas Complejos (GISC),
Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain

<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In the last decades an increasing deal of research has investigated the
phenomenon of opinion misperception in human communities and, more recently,
in social media. Opinion misperception is the wrong evaluation by community’s
members of the real distribution of opinions or beliefs about a given topic.
In this work we explore the mechanisms giving rise to opinion misperception in
social media groups, which are larger than physical ones and have peculiar
topological features. By means of numerical simulations, we suggest that the
structure of connections of such communities plays indeed a role in distorting
the perception of the agents about others’ beliefs, but it is essentially an
indirect effect. Moreover, we show that the main ingredient that generates the
misperception is a spiral of silence induced by few, well connected and
charismatic agents, which rapidly drives the majority of individuals to stay
silent without disclosing their true belief, leading minoritarian opinions to
appear more widespread throughout the community.

_K_ eywords Misperception $\cdot$ Silencing process $\cdot$ Social Simulations
$\cdot$ Opinion Dynamics $\cdot$ Online social interactions

##  1 Introduction

Adherence to group norms and group acceptance are among the most important
factors shaping social behavior in humans. The tendency of the individuals to
conform to the behaviors, opinions and social norms of the others is an innate
trait of human beings, which sometimes can be in contradiction with the
intimate beliefs of the individuals. This interplay between individual and
social environment can have deep consequences on how humans communicate and
transmit information [1] and importantly, on how opinions form and spread
within groups [2, 3].

People’s decisions are often guided by what individuals anticipate the
majority does or approves of [4, 5, 6]. Importantly, these expectations does
not need to be accurate to influence behavior: biases may arise when inferring
the beliefs of others and still guide decision-making. Extensive evidence from
social psychology demonstrates the existence of such biases and their
behavioral outcomes. For instance, individuals may misperceive others’ beliefs
due to false consensus bias, which is the tendency to attribute one’s own
belief to others [7, 8]. Such a bias can detrimentally affect public opinion
processes by facilitating polarization and radicalization phenomena [9].
Unpopular views may also become widespread due to pluralistic ignorance
effects, a situation in which the majority of members privately disagree with
a certain opinion or attitude, and yet publicly conform to it under the wrong
belief that most others accept it [10, 11]. Similar outcomes can also be
induced by few "contrarians" loudly expressing their minoritarian opinions and
by spiralling mechanisms that lead a disagreeing majority into silence [12].
Despite evidences of the effect of misperceptions in supporting unpopular
behaviors and opinions in various real world settings [11, 13, 14, 15, 16,
17], we still lack a clear understanding of how social biases emerge and
impact opinion expression in online social systems. Indeed, by enabling a few
people to easily broadcast their opinions to millions of others [18, 19],
while simultaneously increasing users’ exposure to similar views [20, 21]
social media may provide environments that systematically distort the
inferences users make about the prevalence of certain opinions within their
community, with critical consequences in terms of behavior at collective level
[22, 23, 24].

Several findings support the role of vocal minorities in online debates [25,
26, 27] and of network structures in favoring the emergence of social biases.
For example, a recent study analyzing the communication network structure of
millions of COVID-related fake-news on Twitter provides evidence of an
overexposure of a large group of passive users to the tweets of a few active
users responsible for fabricating most of the contents available online. A
limited understanding of the origin of the information may increase the chance
that misinformation spreads due to users conforming to the "voice of few",
wrongly perceived as representative of the "opinion of many" [28].
Observational data of online behavior however do not allow an in-depth
analysis of the mechanisms responsible for the change in behavior, nor to
identify the potential discrepancy between what individuals publicly declare
and what they privately think. Yet, unfolding the complexity of such dynamics
empirically is not trivial [29, 30]. In this respect, agent-based modelling
(ABM) may represent a useful approach, as it allows building simplified models
of the dynamic under investigation along with an a-priori definition of key
variables (e.g., agents’ decision-making rules) and system parameters (e.g.,
network structure) and to observe which macroscopic patterns can spontaneously
emerge [31]. Specifically, ABM consists in simulating social dynamics by means
of virtual agents that interact among themselves following some established
rules [32, 33, 34]. Recent effort in ABM has indeed been addressed the
modelling of the dynamics of opinion expression, in particular the spiral of
silence, by incorporating the potential for a disconnection between agents’
expressed and private opinions [35, 36] and, in some cases, network
heterogeneity [35, 37, 38]. While in principle these are adequate models to
study opinion dynamics for specific scenarios (e.g., elections with major
candidates, where binary options are equally likely to occur) they are less
suited to examine the processes by which a small fraction of individuals
holding extreme positions (hereby "contrarians") can fuel opinion
misperception in majority holding more moderate views and sway the public
discourse toward their side. Simulations have also been used to explore
whether perceptual social biases can emerge simply due to the geometry of the
network, showing how simple topological effects can increase the perceived
prevalence of few high-degree nodes in the eyes of their less connected
neighbors: in the context of online systems this "illusion of majority" bias,
rooted in the network structure, has indeed been suggested to act as an
exogenous mechanism facilitating the spread of unpopular attitudes and
behaviors ([39]). Yet, this crucial point requires still to be empirically
tested. Further simulation work that have modelled social biases as stemming
from a wrong inference of others’ beliefs have shown how "endogenous" sources
of misperception (e.g., pluralistic ignorance mechanisms) play a key role in
explaining how unpopular norms can spread in a system [40]. Models of this
kind could also be applied to investigate how minorities can influence opinion
expression in systems characterized by a heterogeneous (e.g., majority-
minority) distribution of beliefs.

New models and theoretical approaches that account for both exogenous and
endogenous determinants of opinion misperception are thus necessary to better
understand the effect of social biases on the dynamics of opinion expression
in complex communication networks. In this study we will thus address such a
challenge by modelling communication among agents in a prototypical online
community scenario: when a committed minority of users holding contrarians
beliefs ( "contrarians") enters a population where some belief is commonly
accepted though in diverse grade. For example, we can think about the exordia
of the antivax movement, when few activists started to spread in internet
their skepticism among communities which until have never questioned
vaccinations [41, 42]. By systematically varying Kc, the degree of nodes on
which contrarians are placed, we aim to explore if (and so, how) minoritarian
agents embedded in a network with social-media like features can ignite a
change in the perceived public opinion and the repercussion of such
misperception at the system level.

##  2 Exogenous contributions to opinion misperception

In this section we explore whether a substantial distortion in the opinion
distribution (as perceived by agents) can emerge mainly due to the complexity
of the network, without resorting to the cognitive features of the agents and
their interaction. More precisely, it has been pointed out that in complex
networks the most connected nodes may be over-represented in the neighbourhood
of other individuals, and this may induce agents to misperceive the actual
opinion distribution: it is the so-called "frienship paradox" [43, 44, 45],
which states that in a heterogeneous topology the average degree of the
neighbours of an agent is higher than the network’s average degree itself.
Building on previous modelling work [39] we thus create a model where agents
have minimal cognitive features and express their opinions in a heterogeneous
communication network characterized by a clear majority-minority belief
distribution. By manipulating the popularity of committed minoritarian agents
we aim to evaluate the weight of topological heterogeneity in driving the
dynamics of opinion misperception at macro-scale level.

###  2.1 The Model

We consider a system of $N$ agents arranged on a given network, defined by its
adjacency matrix $\hat{M}_{ij}$ and characterized by its degree distribution
$P(x)$ [44]. As already stated, the internal structure of the agents and their
interaction rules will be outlined in the simplest possible way. Specifically,
each agent $i$ is defined as follows:

  * •

The belief, or private opinion, $b_{i}$, constant in time, which can assume
one of two possible values, $b_{i}=\pm 1$;

  * •

The strength of the opinion $\sigma_{i}\in(0,1]$, also constant in time, which
specifies how strongly the agent holds its belief: when requested to publicly
declare its opinion, the agent will answer its true belief with probability
$\sigma_{i}$: this is the only variable describing the internal state of the
agents;

  * •

The declared opinion $\omega_{i}$, which is what the agent tells the others
about its own belief; it may or may not be equal to the private belief
depending on the probability $\sigma_{i}$. At the initial stages of the
dynamics, it may also assume the $0$ value besides $\pm 1$ (see below for
clarifications).

Once set the agents in the network and assigned the initial values of each
variable (see below), the dynamics takes place. The elementary time-step of
the dynamics is defined as follows:

  * •

An agent $i$ is selected at random;

  * •

With probability $\sigma_{i}$, the agent will declare its private opinion,
therefore $\omega_{i}=b_{i}$;

  * •

With probability $1-\sigma_{i}$ the agent will declare the opinion expressed
by the majority of its own neighbours:

$\omega_{i}=\text{sign}\left[\sum_{j\in\Gamma_{i}}\omega_{j}\right]\ ,$

where $\Gamma_{i}$ is the set of the nearest neighbours of the agent $i$ (if
the result is zero the expressed opinion will be $+1$ or $-1$ with equal
probability);

  * •

A time unit of the simulation will be given by $N$ of these elementary
processes;

  * •

In order to have enough statistics, all the results presented in the following
are averaged over 10000 independent realizations.

Agents thus undergo a basic social influence process: the estimation of the
majority opinion can only be based on what they can observe, i.e., the
opinions expressed by their neighbours. Neither the global distribution of
expressed opinion nor anyone else’s private beliefs is known. The adoption of
such a heuristic is justified under a bounded rationality view of decision
making, e.g., agents acting under limited information and limited capacity to
make decisions. Therefore, the majoritarian opinion expressed by agents’
neighbors is what agents can use to infer the "opinion climate", and will thus
also correspond to agent’s perception of the global distribution of private
beliefs.

Topology: we considered $N=5000$ agents on a Scale Free (SF) network with
degree distribution with exponent $\lambda=2.2$, which characterizes with a
good approximation the topological structure of many social media communities
[43, 46, 47, 48], and in particular on Twitter [49]. Therefore, we consider a
network whose degree distribution follows the distribution below:

$P(k)\propto k^{-\lambda}\ ,$ (1)

where $\lambda=2.2$. We generated such networks for our simulations by means
of the Molloy-Reed algorithm [50], so that for numerical reasons the minimum
allowed degree is 2, the maximum $\lfloor\sqrt{N}\rfloor$ [51].

State variables: the first important variable describing the state of the
system is $\langle\omega\rangle$, that is, the average declared opinion:

$\langle\omega\rangle=\langle\omega\rangle(t)\equiv\frac{\sum_{i=1}^{N}\omega_{i}(t)}{N}\
.$

Analogously, we consider the average private belief $b$, which is defined in
the same way as the average declaration. For simplicity, in our model beliefs
are not allowed to change over time. In order to evaluate the extent to which
network features contribute to bias agents’ perception of others’ opinions we
also include a variable measuring the opinion climate that an agent perceives
in its immediate surrounding. In order to define such variable, we start from
the neighbors’ average declaration $\Delta_{i}$ of each agent:

$\Delta_{i}=\frac{\sum_{j\in\Gamma_{i}}\omega_{j}}{|\Gamma_{i}|}\ ,$

where $|\Gamma_{i}|$ represents the degree of the node $i$. Therefore, we
define the average local opinion climate $r(t)$ as

$r(t)\equiv\langle\Delta_{i}(t)\rangle_{i}=\frac{\sum_{i=1}^{N}\Delta_{i}(t)}{N}\
.$

Notice that all these quantities are defined in the real interval $[-1,+1]$.

As said, agents can only observe what is publicly expressed in their immediate
surrounding, which may or may not be representative of the global
distribution, and may or may not be aligned with the distribution of private
beliefs. The potential discrepancy between the average opinion expressed
around them (local opinion climate, which in our model is defined as the
perception of the global belief distribution) and what agents declare on
average overall (that is, the global opinion climate) will capture the
contribution of topology to opinion misperception. Similarly, we can use the
mismatch between the average opinion expressed around them and the true
distribution of private, non observable beliefs to measure the full extent of
misperception. In Table 1 we provide a summary of the meaning of the main
variables used to describe the state of the system.

Quantity | Definition
---|---
$\langle b\rangle$ | Average private opinion (belief)
$\langle\omega\rangle$ | Average publicly declared opinion
$\Delta_{i}$ | Declared opinion averaged over agent’ $i$ neighbours
$r(t)=\langle\Delta_{i}\rangle$ | Local opinion climate $\equiv$ Average perception
$\langle b\rangle-\langle\omega\rangle$ | Private vs Public Mismatch (between beliefs and declarations)
$\langle b\rangle-r(t)$ | Misperception
$r(t)-\langle\omega\rangle$ | Topological contribution to misperception
Table 1:  List of the quantities utilized for the system’s description.

Initial conditions: The initial settings are chosen in order to mimic a
prototypical social media environment with few committed users holding
unpopular opinions ("contrarians") and a majority of other users holding more
moderate views. We thus assign to contrarians a belief $b=-1$ and maximum
strength, (i.e., $\sigma=1$) that is, they always express opinions in line
with their private belief. The behavior of these agents also recalls the so-
called "hard cores" in the definition by Noelle-Neumann and Matthes &
colleagues [12, 52], that is, people who stick to their opinion regardless of
the prevailing opinion climate. Crucially, contrarians are selectively placed
on nodes with degrees $K_{c}$, in order to highlight the role of the network’s
topology. In every other node with degree $k\neq K_{c}$ agents will hold
belief $b=+1$ and strength $\sigma$ picked up from a uniform random
distribution between 0 and 1. It follows that the distribution of contrarians
will be given by the same $P(k)$ given in Eq. (1), that is, $P(K_{c})\propto
K_{c}^{-\lambda}$. Specifically, when $K_{c}$ is high the network will be
composed by less contrarians, but very well connected. Contrarily, when
$K_{c}$ is low there will be relatively more abundant but less connected
contrarians in the population. We will thus study how the model behave by
systematically varying $K_{c}$, the degree of nodes (and so their "prestige")
on which contrarians are placed. Once set the initial distribution
$\\{b_{i}\\}$, the first time an agent is required to declare its opinion, if
no one of its neighbours has already declared, with probability $1-\sigma_{i}$
the agent will declare a null opinion: $\omega_{i}=0$.

###  2.2 Results

In Fig. 1 (left) we show the temporal evolution of $\langle\omega\rangle,\
r(t)$ and $\langle b\rangle$ for two different exemplifying values of
contrarians’ degree, $K_{c}=3,\ 50$ scenarios. For $K_{c}=3$ contrarians are
relatively abundant ($\sim 18\%$) but also poorly connected, while for the
$K_{c}=50$ scenario contrarians are extremely few ($<0.001\%$) but also
extremely powerful. In both scenarios, after an initial transient phase, both
$\langle\omega\rangle$ and $r(t)$ reach a final stable value ($\langle
b\rangle$ is by definition constant in time). Agents’ perception of others
opinion is always lower than the opinion expressed on overage by agents
($\langle\omega\rangle$), suggesting an over-exposure to contrarians’ opinion
in the local neighbourhood. However, such a distortion is rather small (about
$5\%$). When taking into account the actual distribution of private beliefs
$\langle b\rangle$, the discrepancy becomes considerably larger. This is
easily observable in the $K_{c}=3$ scenario (the behavior for $K_{c}=50$ is
qualitatively the same): agents observe an average opinion around them of
slightly less than 0.2, while the average private belief is around 0.67.
Overall, the behavior of this model suggests that topology itself contributes
only partially to the opinion misperception and that most of the phenomenon is
due to the mismatch between what agents publicly declare and their private
beliefs. Such a pattern implies that even when agents are modelled with
minimal cognitive ingredients (i.e., strength of belief), the main source of
bias appears to be endogenous (that is, mainly due to agents’ behavior) rather
than exogenous (that is, their position in the network). Fig. 1 (right) shows
the final state of $\langle b\rangle,\ \langle\omega\rangle$ and $r$ as
functions of the contrarians’ degree. As can be observed, this same conclusion
holds for every value of $K_{c}$ . With increasing $K_{c}$ all the curves tend
to 1 because the size of the contrarians minority becomes drastically small,
giving rise to a saturation effect.

Figure 1: Left: Time evolution of the average declared opinion
$\langle\omega\rangle(t)$ (continuous lines), local opinion climate $r(t)$
(dotted lines), and average private belief $\langle b\rangle$ (dashed lines,
constant in time) for two exemplifying scenarios: $K_{c}=3,\ 50$, in black and
violet respectively. Scale-free network, size $N=5000$, $\lambda=2.2$,
averages over 10000 independent realizations. We highlighted the contributions
to misperception given by the topological features of the system (in yellow),
i.e. network heterogeneity, and the endogenous one of the agents (in blue),
i.e., the discrepancy between what agents believe and what publicly declare.
Right:  Final values of the average declaration $\langle\omega_{f}\rangle$
(black line), average local opinion $\langle r_{f}\rangle$ (green line), and
average belief $\langle b\rangle$ (red lines) as functions of contrarians’
degree $K_{c}$. Scale-free network, size $N=5000$, $\lambda=2.2$, averages
over 10000 independent realizations. We highlighted again the contribution to
misperception given by the topological features (yellow double arrow) and the
one "endogenous" (i.e., due to the cognitive features of the individuals) of
the agents, i.e., discrepancy between private vs publicly declared agents’
opinion (blue double arrow).

##  3 Endogenous contributions to misperception

The results obtained in the previous section show that topological effects
alone are not enough to explain the overall misperception of the opinions held
among the population, without resorting to more complex cognitive features of
the agents. In order to better understand what drives the observed dynamics,
we thus proceed by refining the characterization of the endogenous component
of opinion misperception by acting on agents’ interaction rules. Implementing
complex agents’ socio-cognitive features in ABM is a non-trivial task [53],
mainly because of the limits of encompassing the complexity of human cognition
in a relatively simple algorithm. There are many ways to model the internal
dynamics of agents, as for example refining models from Opinion Dynamics
theory [54]. A useful example comes from the study of Merdes [40] which
investigated the role of belief misperception in the emergence of unpopular
norms by means of simulations. In that model, agents have fixed, non
observable, beliefs and are subject to social pressure from the surrounding
neighborhood with regard to which behavior to adopt (e.g., to follow or not
the norm by conforming to the most prevalent behavior). Agents’ choice is
driven by the effort to minimize the gap between their social expectations –
what agents believe most others do (or perceive so) – and their own,
potentially disagreeing, private belief. Such a characterization of agents’
decision rule, along with the possibility for them to behave differently from
their private beliefs creates the conditions for the emergence of
misperception at macro-scale level, i.e., most agents behave in a way which is
not aligned with what would be expected from the underlying distribution of
private preferences in the population. Norms can be thought as a specific kind
of opinion, and our differentiation between agents’ private and publicly
declared opinion (observable choice) allows us to adapt their formulation of
social influence to describe the endogenous contribution to opinion
misperception. Accordingly, in our model the social component will be
formulated as the effort by the agents to minimize the gap between what they
perceive is the majoritarian opinion around them and their own private – and
potentially disagreeing – opinion. As in [40], such assumption should be able
to capture human tendency to reduce as possible the deviations from others’
behaviors, norms and opinions [5]. Crucially, people’s perception of their
social surrounding (whether in line or not with their private opinion) may not
only affect which opinion to declare, but also, whether to declare at all.
More precisely, the perceived disagreement with the opinion mostly declared in
the agents’ neighborhood is expected to limit agents likelihood to express
their own private belief [12] and may thus further contribute to the emergence
of opinion misperception in the system. Accordingly, we incorporate this
insight by adding the possibility for agents to stay silent.

###  3.1 The Model

In order to describe the above processes mathematically, a specific goal
function $G_{i}$ is introduced for every agent. This function gives the payoff
gained by the agent according to its own and neighbors’ state, so that the
payoff increases when the state of the agent is closer to the others’, but
decreases when the agent is forced to change its own previous state. In other
words, agents tend to minimize the disagreement with their neighbours’
opinions as well as the cost to act or declare in contrast with ones’ own
private belief. More precisely, we assume that agents have a private opinion
$b_{i}=\pm 1$, constant in time, and an expressed opinion $\omega_{i}=0,\pm
1$, where $\omega_{i}=0$ critically adds a preference for staying silent111In
the first model, the option $\omega_{i}=0$ can be adopted only initially, when
the agent does not declare its belief but no neighbor has still declared
anything: therefore, it is not meant to represent agents’ choice to stay
silent.. Again, to each agent $i$ is attached a belief’s strength
$\sigma_{i}$, which can assume a real value from 0 (no strength at all), to 1
(maximum of confidence in own belief). The dynamics in this case proceeds as
follows:

  * •

An agent $i$ is selected at random;

  * •

To decide what to explicitly declare or whether to remain silent, the agent
$i$ maximizes the goal function $G(\omega_{i})$ defined as

$G(\omega_{i})\equiv-\left|\frac{\sigma_{i}b_{i}+\sum_{j\in\Gamma_{i}}\mathcal{A}_{i}^{j}\
\omega_{j}}{|\Gamma_{i}|+1}-\omega_{i}\right|\ ,$ (2)

where $|\Gamma_{i}|\equiv\sum_{j\in\Gamma_{i}}|\omega_{j}|$, $\sigma_{i}$ is
the strength of the belief of player $i$, and the function
$\mathcal{A}_{i}^{j}$ is the "prestige" of the agent $j$ with respect to agent
$i$ itself: we set $\mathcal{A}_{i}^{j}=\frac{\deg(j)}{\deg(i)}$ to
incorporate the fact that more charismatic individuals tend to get more
connections with others than less charismatic ones [55, 56];

  * •

Once found the value $\bar{\omega}_{i}$ which maximizes $G(\omega_{i})$, such
$\bar{\omega}_{i}$ will be the declaration choice (opinion expressed) by $i$
in the next interaction (if $\bar{\omega}_{i}=0$, agent $i$ stays silent).

  * •

Notice that one could equivalently define the cost function
$C(\omega_{i})\equiv-G(\omega_{i})$, and proceed in the same way with the only
difference being that $C$ has to be minimized.

Topology: As for the first model, we analyze Scale-Free networks with degree
distribution $P_{sf}(k)\propto k^{-\lambda}$ with $\lambda=2.2$, as in Eq.
(1), generated again by means of a Molloy-Reed algorithm (see Sec. 2); we also
accomplished some simulations with $\lambda=1.5$ in order to test the effect
of increasing heterogeneity.

State variables:  To quantitatively characterize the model’s behavior, we use
the same variables defined in Subsec. 2.1, that is, the average declared
opinion $\langle\omega\rangle$ and the average perception $r$ through the
local opinion perception $\Delta_{i}$.

Initial conditions: as for the first model, we consider systems with most
agents having $b_{i}=1$ and few agents with negative opinions put on nodes of
degree $K_{c}$; we set $\sigma_{i}=1$ for the latter agents, and uniformly
distribute it in the interval $[0,1]$ for the former ones. Finally, we always
assume $\omega_{i}=0\ \forall i$ at the very beginning of the dynamics.

As in the previous section, we will study how the model behaves by
systematically varying $K_{c}$, the degree of nodes on which contrarians are
placed, and all the results presented in the following section are averaged
over thousands independent realizations.

###  3.2 Results

In Fig. 2 (left) we show the temporal evolution of agents’ average declaration
$\langle\omega\rangle$ for systems on SF networks ($\lambda=2.2,\ N=5000$),
for three different exemplifying $K_{c}$ scenarios. Again, increasing $K_{c}$
values correspond to systems with fewer but better connected (and more
"charismatic") contrarians.

Figure 2: Left: Temporal evolution of the average declaration for systems on
SF networks with $\lambda=2.2$, and different values of $K_{c}$. System size
$N=5000$, averages over 5000 independent realizations. Right: behavior of the
final average belief, local perception and declaration as functions of the
contrarians’ degree $K_{c}$ for a highly heterogeneous SF networks,
$\lambda=2.2$. System size $N=5000$, results averaged over 5000 independent
realizations.

The average opinion expressed by agents is quite close to 1 at the initial
stages of the dynamics (despite the fact that $\omega_{i}(t=0)\equiv 0\
\forall i$, the agents declare their true belief the first time they express
themselves), but decreases up to a relatively lower final value with similar
qualitative behavior for different contrarians’ degree. Also, the final
declaration values are higher with increasing contrarians degree value, as the
overall size of contrarians in the systems becomes drastically small (e.g., at
$K_{c}$ =20 only 3 out of 1000 agents are contrarians), but with a saturation
effect for larger $K_{c}$.

In Fig. 2 (right) we show the final values of the average belief, local
opinion climate and the actual opinion declaration, as a function of
contrarians’ degree $K_{c}$. The average belief value (which is constant in
time) increases with $K_{c}$ because the number of contrarians decreases
according to the increase in their degree. However, the average opinion
declared by agents, both as it is perceived locally as well as it is declared
globally, is always lower than the belief value in the population, and a
stable plateau is reached with increasing $K_{c}$ values. This pattern is very
different from the results of first model, where the average declaration value
approached agents’ belief values as $K_{c}$ increases (see Fig. 1 (right)).
Such a finding suggests that, even when being of a negligible number, well-
connected contrarians are able to sway the global opinion towards their side,
despite the fact that almost every agent holds a disagreeing private belief.
In order to better understand the observed pattern, we look at how the
proportions of agents choosing to declare the majoritarian or minoritarian
opinion (or to stay silent) change according to $K_{c}$.

Figure 3: Frequency of the declared opinions, perception and beliefs in the
system final state for different degree $K_{c}$ of the initial contrarians,
distinguished among the different components of the population (majority,
minority, silenced) are shown. Scale-free networks, size $N=5000$,
$\lambda=2.2$.

Fig. 3 shows the proportion of agents based on their final declaration choices
$\omega$ at the end of dynamics for several values of $K_{c}$ from to 20 as
compared to the fixed distribution of their private beliefs. The figure
suggests that the shift towards the contrarian opinion is not due to
majoritarian agents publicly aligning to contrarians’ views, but to majority’s
choice of staying silent. Indeed, silence becomes the most effective solution
for agents to minimize the cost of having a different belief from their (even
though misperceived) immediate social surrounding without the need to deviate
from their inner views. From a dynamical point of view, the possibility to
avoid explicit declarations induces a cascade where agents surrounded by
neighbours declaring mostly the opposite belief or being silent end up staying
silent themselves, driving others to stay silent on their turn, and so on.
Such a dynamics leads to a final state where the average declared opinion is
much more skewed toward the opinion of contrarians compared to the average
group’s private beliefs. Interestingly, around $55\div 60\%$ of subjects
(almost everyone of which holding positive beliefs) end up being silent,
regardless of contrarians’ degree. In order to shed light on this effect, we
compute the number of silenced agents per contrarian and plot it as a function
of contrarians’ degree. Fig 4 shows a superlinear relationship between the
contrarians’ degree value and the amount of majoritarian agents silenced.
Specifically, when $K_{c}$ is low, e.g., 2, the size of the minority is large
(around $35\%$), but every minoritarian agent drives less than 2 agents to
silence. This suggests that for low $K_{c}$ silence mostly arises due to
short-range dynamics, e.g., many contrarians directly influencing their
immediate neighbors, and very few others. At $K_{c}=20$, the minority is
negligible ($<0.1\%$), but for every minoritarian agent there are $\sim 150$
majoritarian agents that get silenced, much more than their neighbours. Such a
pattern suggests the presence of long-range dynamics, as silence spreads even
beyond contrarians’ direct connections. Therefore, for high $K_{c}$ the
increasing prestige of well-connected contrarians balances the reduction in
their number, inducing the saturation effect that stabilizes the global
declaration (and perception of local, too) toward lower values. Such a
saturation effect is also reflected in the exponent of the superlinear
relationship between contrarians and silenced agents, which is very close to
the network $\lambda$ value: since the overall fraction of silent individuals
remains approximately constant while the number of contrarians decrease as
$K_{c}^{-\lambda}$, then the amount of the former versus the latter must
increase with an exponent very close to $\lambda$ itself.

Figure 4: Black circles: number of silenced agents per contrarian as a
function of the contrarians’ degree $K_{c}$. Dashed line:  power-law fit $\sim
K_{c}^{\nu},\ \nu\simeq 2.1$ (superlinear behavior). Red/blue lines:
difference between the number of agents silenced per contrarian and the actual
number of contrarians’ nearest neighbors. For $K_{c}=2$ the line is red
because each contrarian drives less agents to silence than its number of
nearest neighbors. System parameters: SF network with exponent $\lambda=2.2$,
system size $N=5000$; averages over 5000 independent realizations.

As in the first model, the direct effect of topology in distorting agents’
perception of others’ opinion is again rather small. Interestingly, and in
contrast to the former, the opinion that users can observe locally is actually
closer to the true value of private beliefs as compared to the opinion
declared on average by agents. Such an effect is possibly due to the
stochasticity of the model, e.g., by few agents holding positive beliefs
randomly ending on nodes with degree values even higher than the ones of
contrarians. Such agents are thus able to limit the "traction" of contrarians
with high prestige, whose effect is therefore top-down. Such a finding
suggests that the direct topological effect on opinion misperception is,
besides small, also non-universal. Finally, we replicated the model on a
Scale-free network with $\lambda=1.5$, corresponding to a more heterogeneous
system than a Twitter-like one. From a qualitative point of view, a similar
behavior can be observed (see Fig. 5), confirming that the results obtained
are stable in the framework of SF networks.

Figure 5: Behavior of the final average belief, local opinion and declaration
as functions of the contrarians’ degree $K_{c}$ for a highly heterogeneous SF
networks, $\lambda=1.5$. System size $N=5000$, results averaged over 5000
independent realizations.

##  4 Discussion

There is increasing evidence that social media communication features increase
the potential for committed minorities to reach a much larger audience than in
offline contexts, and to easily give rise to viral phenomena [57, 58]. While
in principle such effect may not be deleterious per se, an over-representation
of otherwise minority opinions may become problematic if it increases the risk
of inflating the support for socially questionable or even harmful views [28,
35]. This study thus aims at better understanding how opinion misperception
emerges as a results of agents dynamically interacting in networks with
social-media like features. Specifically, we focused on evaluating the
contribution of both exogenous (e.g., the structure of connections) and
endogenous (e.g., agents’ socio-cognitive features) mechanisms allowing
committed minorities to be over-represented in the eyes of other users and the
consequences of such distortion on the dynamics of opinion expression at
collective level. Topological contribution was analyzed by selectively placing
contrarians on nodes with degree $K_{c}$ and by running simulations for
different degree $K_{c}$ values. Simulation results show that local network
effects (in particular, the heterogeneity of the degree distribution) only
marginally distort the perceived prevalence of minority opinions with respect
to what agents declare on average. Besides being marginal, the effect of such
topologically-induced bias in social perception also does not seem to be
universal. Indeed, under some conditions the perceived prevalence of
contrarians opinions in the local surrounding can even get closer to the
actual distribution of agents’ private beliefs, thus partially correcting the
effects of opinion misperception on the system. Previous modelling effort has
shown how "illusion of majority" effects arising simply due to network
features [39] can increase the perceived prevalence of minority opinions and
act as network based explanation for the emergence of social biases. However,
the actual impact of such distortions on collective behavior remains untested.
By allowing agents to dynamically interact and by incorporating a
differentiation between public vs private opinion we can explicitly test the
effect of such distortions on public opinion. Our results suggest that
behavior is driven by agents conforming under a wrong inference of
contrarians’ beliefs, which eventually shifts the global public opinion toward
contrarians side. Crucially, the result of the second model suggests that this
effect is not due to a majority choosing to align with the contrarians
minority view, but to a majority of agents choosing to stay silent.

A spiral of silence emerges as a consequence of such individual choices.
Silence becomes thus a crucial element to explain the observed opinion dynamic
patterns. While similar mechanisms have already been noticed and discussed in
literature [12, 59, 60], their existence in social media environments has been
subject of debate [59, 61, 62, 63]. Observing spiralling processes and their
effects in social media is difficult, mainly because only the views of those
who express their opinions are evident there. Such a barrier can be lifted by
means of computer simulations that allow to explicitly model the discrepancies
between private and publicly expressed opinions. Although based on a
simplistic description of opinion formation, our work suggests that the main
feature of social media networks (i.e., the heterogeneity of connections) can
promote social biases and contribute to amplify the voice of a motivated few
by pushing into silence a disagreeing majority. Surprisingly, the silencing
process emerges regardless of the network position or size of contrarians
minority. The present work underlines the importance of network
characteristics when it comes to observing the dynamical process. Indeed, we
show that when contrarians are located on low-degree nodes (with fewer direct
connections), their lack in individual prestige is compensated by their larger
number, which allow silencing process still to emerge at large scale. On the
other hand, when placed on high-degree nodes, their numbers are few, but
silence emerges due to their strong connections and high prestige, thus
compensating for the significant reduction in their numbers. Consequently,
these two effects balance each other, resulting in a consistently high
fraction of silenced individuals. These findings align with the research
conducted by Ross and colleagues [35], who explored the impact of social bots
(artificial, automated accounts impersonating humans) on diverting public
discussions in social networks using a different model. Similarly to our
findings, they show that the influence of these contrarians does not seem to
be affected significantly by changes in their position or overall number.
Notably, in both studies the percentage of silenced individuals reached levels
around $50\%$ to $60\%$. Despite differences in the models and simulation
algorithms employed, this agreement suggests that the presented findings may
reflect an inherent characteristic of these types of social interactions.

As a consequence of the silencing process the proportion of agents expressing
majority and minority opinions changes considerably with respect to the true
distribution of private beliefs. However, in most of the investigated
scenarios this process does not translate into a complete overturn of the
majoritarian view. If we think of opinion formation and spread in social
media, this suggests that in case of widely accepted views, few influential
contrarians can weaken the strength of majoritarian position but this may not
be enough to make them dominate public debate, as it has already been pointed
out by Acerbi et al. [64, 65]. In our model, the only case of inversion in
majority-minority proportions is found when the minority exceeds the
percentage of $30\%$. Interestingly, such a percentage is close to what
indicated by Centola and Baronchelli [66] as being a tipping point beyond
which a minority manages to spread globally into a population of agents.
Although our study does not allow (nor it was meant to) to investigate the
effects of tipping points on the dynamics of opinion expression, further work
could explore the link, i.e., whether opinion misperception can facilitate or
hinder the emergence of social tipping points, and the role played by the
network itself (e.g., the number and prestige of the agents involved).

In summary, our work contributes to the current literature on opinion dynamics
in complex networks by exploring the determinants of opinion misperception in
heterogeneous network systems. We tackle the issue by incorporating insights
from socio-psychological literature to model the discrepancy between expressed
and private opinions and the resulting effects at large-scale level. An open
question is whether the results of our model can be translatable into “real”
online scenarios: in order to address it, the model should be validated with
empirical data. In related fields of research, experiments have shown that
reducing the perceived prevalence of inappropriate behaviors in online fora
(e.g., frequency of hate speech) significantly affects people willingness to
engage in such actions by changing what people infer from the social context
in terms of social acceptability (e.g., descriptive norms) [67]. Similar
experimental settings could be designed to understand how changes in the
(misperceived) descriptive norms may affect the likelihood that those with
opposing views would speak out. Data of this kind, along with observation of
real networks and survey data could all provide valuable insights and
empirical evidence to validate and expand our understanding of the processes
behind opinion formation, misperception, and their dynamics at different
levels. Finally, it has been shown that the effect of misperceptions can be
alleviated by informing people about the real distribution of others’ beliefs
[68]. If, as suggested here and elsewhere [28], such biases occur in
communication networks too, simulation models could be further used to test
interventions to counteract the effects of such social biases, for example by
testing whether revealing over-exposure to minority opinions can limit the
emergence of silencing processes at system level. Multidisciplinary research
combining experimental social science and computational methods can prove to
be effective in enhancing our comprehension of the complex dynamics at play in
online social interactions and help guide future research and interventions in
this domain.

#### Acknowledgements

This work was partially supported by project SERICS (PE00000014) under the MUR
National Recovery and Resilience Plan funded by the European Union -
NextGenerationEU and by the EU H2020 ICT48 project „Humane AI Net“ under
contract n 952026.

## References

  * [1] Joe Henrich and Robert Boyd.  The evolution of conformist transmission and the emergence of between-group differences.  Evolution and Human Behavior, 19(4):215–241, 1998.
  * [2] Claudio Castellano, Santo Fortunato, and Vittorio Loreto.  Statistical physics of social dynamics.  Reviews of modern physics, 81(2):591, 2009.
  * [3] Parongama Sen and Bikas K Chakrabarti.  Sociophysics: an introduction.  Oxford University Press, 2014.
  * [4] Robert B Cialdini and Melanie R Trost.  Social influence: Social norms, conformity and compliance.  1998\.
  * [5] Cristina Bicchieri.  The grammar of society: The nature and dynamics of social norms.  Cambridge University Press, 2005.
  * [6] Elinor Ostrom.  Collective action and the evolution of social norms.  Journal of economic perspectives, 14(3):137–158, 2000.
  * [7] Lee Ross, David Greene, and Pamela House.  The “false consensus effect”: An egocentric bias in social perception and attribution processes.  Journal of experimental social psychology, 13(3):279–301, 1977\.
  * [8] Joachim Krueger and Russell W Clement.  The truly false consensus effect: an ineradicable and egocentric bias in social perception.  Journal of personality and social psychology, 67(4):596, 1994.
  * [9] Magdalena Wojcieszak.  False consensus goes online: Impact of ideologically homogeneous groups on false consensus.  Public Opinion Quarterly, 72(4):781–791, 2008.
  * [10] Dale T Miller and Cathy McFarland.  Pluralistic ignorance: When similarity is interpreted as dissimilarity.  Journal of Personality and social Psychology, 53(2):298, 1987.
  * [11] Deborah A Prentice and Dale T Miller.  Pluralistic ignorance and the perpetuation of social norms by unwitting actors.  In Advances in experimental social psychology, volume 28, pages 161–209. Elsevier, 1996.
  * [12] Elisabeth Noelle-Neumann.  The spiral of silence a theory of public opinion.  Journal of communication, 24(2):43–51, 1974.
  * [13] James M Fields and Howard Schuman.  Public beliefs about the beliefs of the public.  Public Opinion Quarterly, 40(4):427–448, 1976.
  * [14] Tracy A Lambert, Arnold S Kahn, and Kevin J Apple.  Pluralistic ignorance and hooking up.  Journal of sex research, 40(2):129–133, 2003.
  * [15] Marlene Sandstrom, Heather Makover, and Maria Bartini.  Social context of bullying: Do misperceptions of group norms influence children’s responses to witnessed episodes?  Social Influence, 8(2-3):196–215, 2013.
  * [16] Jacob TN Young and Frank M Weerman.  Delinquency as a consequence of misperception: Overestimation of friends’ delinquent behavior and mechanisms of social influence.  Social Problems, 60(3):334–356, 2013.
  * [17] Christin L Munsch, Cecilia L Ridgeway, and Joan C Williams.  Pluralistic ignorance and the flexibility bias: Understanding and mitigating flextime and flexplace bias at work.  Work and Occupations, 41(1):40–62, 2014.
  * [18] J Scott Brennen, Felix M Simon, Philip N Howard, and Rasmus Kleis Nielsen.  Types, sources, and claims of COVID-19 misinformation.  PhD thesis, University of Oxford, 2020.
  * [19] Nicole M Krause, Isabelle Freiling, Becca Beets, and Dominique Brossard.  Fact-checking as risk communication: the multi-layered risk of misinformation in times of covid-19.  Journal of Risk Research, 23(7-8):1052–1059, 2020.
  * [20] C Thi Nguyen.  Echo chambers and epistemic bubbles.  Episteme, 17(2):141–161, 2020.
  * [21] Matteo Cinelli, Gianmarco De Francisci Morales, Alessandro Galeazzi, Walter Quattrociocchi, and Michele Starnini.  The echo chamber effect on social media.  Proceedings of the National Academy of Sciences, 118(9):e2023301118, 2021.
  * [22] Alessandro Bessi and Emilio Ferrara.  Social bots distort the 2016 us presidential election online discussion.  First monday, 21(11-7), 2016.
  * [23] Jeffrey Lees and Mina Cikara.  Understanding and combating misperceived polarization.  Philosophical Transactions of the Royal Society B, 376(1822):20200143, 2021.
  * [24] Samantha L Moore-Berg, Lee-Or Ankori-Karlinsky, Boaz Hameiri, and Emile Bruneau.  Exaggerated meta-perceptions predict intergroup hostility between american political partisans.  Proceedings of the National Academy of Sciences, 117(26):14864–14872, 2020.
  * [25] Sandra González-Bailón, Javier Borge-Holthoefer, and Yamir Moreno.  Broadcasters and hidden influentials in online protest diffusion.  American behavioral scientist, 57(7):943–965, 2013.
  * [26] Nir Grinberg, Kenneth Joseph, Lisa Friedland, Briony Swire-Thompson, and David Lazer.  Fake news on twitter during the 2016 us presidential election.  Science, 363(6425):374–378, 2019.
  * [27] Colleen McClain, Regina Widjaya, Gonzalo Rivero, and Aaron Smith.  The behaviors and attitudes of us adults on twitter.  2021\.
  * [28] Piergiorgio Castioni, Giulia Andrighetto, Riccardo Gallotti, Eugenia Polizzi, and Manlio De Domenico.  The voice of few, the opinions of many: evidence of social biases in twitter covid-19 fake news sharing.  Royal Society Open Science, 2022.
  * [29] Robert Axelrod.  The dissemination of culture: A model with local convergence and global polarization.  Journal of conflict resolution, 41(2):203–226, 1997.
  * [30] Michael W Macy and Robert Willer.  From factors to actors: Computational sociology and agent-based modeling.  Annual review of sociology, 28(1):143–166, 2002.
  * [31] Joshua M Epstein.  Agent-based computational models and generative social science.  Complexity, 4(5):41–60, 1999.
  * [32] Rosaria Conte, Bruce Edmonds, Scott Moss, and R Keith Sawyer.  Sociology and social theory in agent based social simulation: A symposium.  Computational & Mathematical Organization Theory, 7(3):183–205, 2001.
  * [33] Rosaria Conte.  Agent-based modeling for understanding social intelligence.  Proceedings of the National Academy of Sciences, 99(suppl_3):7189–7190, 2002.
  * [34] Dirk Helbing.  Agent-based modeling.  Springer, 2012.
  * [35] Björn Ross, Laura Pilz, Benjamin Cabrera, Florian Brachten, German Neubaum, and Stefan Stieglitz.  Are social bots a real threat? an agent-based model of the spiral of silence to analyse the impact of manipulative actors in social networks.  European Journal of Information Systems, 28(4):394–412, 2019.
  * [36] Dongyoung Sohn and Nick Geidner.  Collective dynamics of the spiral of silence: The role of ego-network size.  International Journal of Public Opinion Research, 28(1):25–45, 2016\.
  * [37] Yue Wu, Ya-Jun Du, Xian-Yong Li, and Xiao-Liang Chen.  Exploring the spiral of silence in adjustable social networks.  International Journal of Modern Physics C, 26(11):1550125, 2015\.
  * [38] Siyuan Ma and Hongzhong Zhang.  Opinion expression dynamics in social media chat groups: an integrated quasi-experimental and agent-based model approach.  Complexity, 2021:1–14, 2021.
  * [39] Kristina Lerman, Xiaoran Yan, and Xin-Zeng Wu.  The" majority illusion" in social networks.  PloS one, 11(2):e0147617, 2016.
  * [40] Christoph Merdes.  Growing unpopular norms.  Journal of Artificial Societies and Social Simulation, 20(3), 2017\.
  * [41] Ashwin Rao, Fred Morstatter, Minda Hu, Emily Chen, Keith Burghardt, Emilio Ferrara, Kristina Lerman, et al.  Political partisanship and antiscience attitudes in online discussions about covid-19: Twitter content analysis.  Journal of medical Internet research, 23(6):e26692, 2021.
  * [42] Staci L Benoit and Rachel F Mauldin.  The “anti-vax” movement: a quantitative report on vaccine beliefs and knowledge across social media.  BMC public health, 21(1):1–11, 2021.
  * [43] Guido Caldarelli.  Scale-free networks: complex webs in nature and technology.  Oxford University Press, 2007.
  * [44] Guido Caldarelli and Michele Catanzaro.  Networks: A very short introduction, volume 335.  Oxford University Press, 2012.
  * [45] Scott L Feld.  Why your friends have more friends than you do.  American journal of sociology, 96(6):1464–1477, 1991.
  * [46] W Fan and KH Yeung.  Online social networks—paradise of computer viruses.  Physica A: Statistical Mechanics and its Applications, 390(2):189–197, 2011.
  * [47] Imre Varga.  Scale-free network topologies with clustering similar to online social networks.  In Proceedings of the International Conference on Social Modeling and Simulation, plus Econophysics Colloquium 2014, pages 323–333. Springer, Cham, 2015.
  * [48] Sofía Aparicio, Javier Villazón-Terrazas, and Gonzalo Álvarez.  A model for scale-free networks: application to twitter.  Entropy, 17(8):5848–5867, 2015.
  * [49] Elisa Omodei, Manlio De Domenico, and Alex Arenas.  Characterizing interactions in online social networks during exceptional events.  Frontiers in Physics, 3:59, 2015.
  * [50] Michael Molloy and Bruce Reed.  A critical point for random graphs with a given degree sequence.  Random structures & algorithms, 6(2-3):161–180, 1995.
  * [51] Ivan Kryven.  General expression for the component size distribution in infinite configuration networks.  Physical Review E, 95(5):052303, 2017.
  * [52] Jörg Matthes, Kimberly Rios Morrison, and Christian Schemer.  A spiral of silence for some: Attitude certainty and the expression of political minority opinions.  Communication Research, 37(6):774–800, 2010.
  * [53] Andreas Flache, Michael Mäs, Thomas Feliciani, Edmund Chattoe-Brown, Guillaume Deffuant, Sylvie Huet, and Jan Lorenz.  Models of social influence: Towards the next frontiers.  Journal of Artificial Societies and Social Simulation, 20(4), 2017\.
  * [54] Francesca Giardini, Daniele Vilone, and Rosaria Conte.  Consensus emerging from the bottom-up: the role of cognitive variables in opinion dynamics.  Frontiers in Physics, 3:64, 2015.
  * [55] Juan-Carlos Pastor, James R Meindl, and Margarita C Mayo.  A network effects model of charisma attributions.  Academy of Management Journal, 45(2):410–420, 2002.
  * [56] Benjamin Tur, Jennifer Harstad, and John Antonakis.  Effect of charismatic signaling in social media settings: Evidence from ted and twitter.  The Leadership Quarterly, page 101476, 2021.
  * [57] Marco Guerini, Carlo Strapparava, and Gozde Ozbal.  Exploring text virality in social networks.  In Proceedings of the International AAAI Conference on Web and Social Media, volume 5, pages 506–509, 2011.
  * [58] Gohar Feroz Khan and Sokha Vong.  Virality over youtube: an empirical analysis.  Internet research, 24(5):629–647, 2014.
  * [59] Dietram A Scheufle and Patricia Moy.  Twenty-five years of the spiral of silence: A conceptual review and empirical outlook.  International journal of public opinion research, 12(1):3–28, 2000\.
  * [60] Jörg Matthes.  Observing the “spiral” in the spiral of silence.  International Journal of Public Opinion Research, 27(2):155–176, 2015.
  * [61] Andrew F Hayes and Jörg Matthes.  Self-censorship, the spiral of silence, and contemporary political communication.  2014\.
  * [62] Elihu Katz and Yonatan Fialkoff.  Six concepts in search of retirement.  Annals of the International Communication Association, 41(1):86–91, 2017.
  * [63] Jörg Matthes, Johannes Knoll, and Christian von Sikorski.  The “spiral of silence” revisited: A meta-analysis on the relationship between perceptions of opinion support and political opinion expression.  Communication Research, 45(1):3–33, 2018.
  * [64] Alberto Acerbi.  Cultural evolution in the digital age.  Oxford University Press, 2019.
  * [65] Alberto Acerbi, Sacha Altay, and Hugo Mercier.  Research note: Fighting misinformation or fighting for information?  Harvard Kennedy School Misinformation Review, 2022.
  * [66] Damon Centola, Joshua Becker, Devon Brackbill, and Andrea Baronchelli.  Experimental evidence for tipping points in social convention.  Science, 360(6393):1116–1119, 2018.
  * [67] Amalia Álvarez-Benjumea and Fabian Winter.  Normative change and culture of hate: An experiment in online environments.  European Sociological Review, 34(3):223–237, 2018.
  * [68] Alan D Berkowitz.  An overview of the social norms approach.  Changing the culture of college drinking: A socially situated health communication campaign, 1:193–214, 2005.