Running head: EMERGENCE OF SPIRAL OF SILENCE
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The Emergence of Spiral of Silence from Individual Behaviors:
Agent-based Modeling of Spiral of Silence
Chengjun WANG
Author Note
Wang Chengjun, Department of Media and Communication,
City University of Hong Kong
Correspondence concerning this article should be addressed to Wang Chengjun
Department of Media and Communication, City university of Hong Kong,
Room 105D, Hall 8, 83 Tat Chee Avenue, Kowloon. Hong Kong.
Tel: 852-9644 2905
Fax: 852-3442 0228
E-mail: [email protected]
Biographical Statement:
My name is Chengjun WANG. I‟m a Ph.D student of the department of media &
communication, City University of Hong Kong. Currently, I am also a research member of
web mining lab. I received my master degree from the Beijing University, and Bachelor
degree from LanZhou University. Focusing on news diffusion on social media, I am
interested in statistics, social network analysis, and agent-based models. I am currently
working on modeling public opinion with the perspective of threshold models.
Running head: EMERGENCE OF SPIRAL OF SILENCE
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Abstract
The purpose of this exploratory study is threefold: first, to analytically explore the boundary
conditions of the robust existence of spiral of silence; second, to gauge how social
interactions influence the formation of spiral of silence; third, to analyze the dynamic
characteristics of the process in the aspect of size-dependent and time heterogeneity. By
proposing an agent-based model of spiral of silence, the findings suggest: first, stable
existence of spiral of silence is contingent upon the comparative strength of mass media over
reference groups; second, heterogeneous individuals‟ bottom-up interactions with mass media
and reference groups at the local scope give rise to the spiral of silence as an emergence of
macroscopic regularity, to be specific, the growth rate of spiral of silence decreases over time,
the number of the people falling silent per time is in accordance with Pareto distribution.
Key Words: spiral of silence theory, willingness to express, agent-based model,
interaction, reference group
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The Emergence of Spiral of Silence from the Individual Behavior:
Agent-based Modeling of the Spiral of Silence Theory
The formation and change of public opinion are important in terms of public opinion
research. Spiral of silence theory explains the opinion dynamics in the aspects of individuals‟
interactions with mass media and reference groups.
However, there are many debates on the fundamental requisites of the theory, and the
inconsistent findings of empirical studies on the theory intensify these criticisms (Donsbach
& Traugott, 2007; Glynn, Hayes, & Shanahan, 1997; Scheufle & Moy, 2000). In addition to
that, another criticism is a lack of analyzing the interplay between aggregate-level variables
(such as social settings) and individual-level predictors (Donsbach & Traugott, 2007).
The main research question of this present study derives from the previous studies: first,
what‟re the boundary conditions of the theory? Second, how spiral of silence come out, and
evolves along time?
Spiral of silence theory is not only a macro-theory, but also a dynamic process, and it
links the macro-levels, meso-levels, and the micro-levels of analysis (Donsbach & Traugott,
2007). Using agent-based models to integrate the aggregate level factors (the influences of
the reference groups, and the influence of the mass media) and individual level behaviors
(willingness to express), this study aims to: first, inquire the boundary conditions of the stable
existence of spiral of silence, and second, examine the core hypothesis of the theory, that is,
how individuals‟ interactions with the dual climates of opinion (the reference group and the
mass media) at the local scope give rise to the emergence of the spiral of silence as a global
regularity?
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In the literature review part, I will review the spiral of silence theory, especially the
influence of mass media, and the influences of reference groups.
Literature Review
“Last-minute Swing” and the Origin of Spiral of Silence Theory
Noelle-Neumann (E. Noelle-Neumann, 1974; 1993) is puzzled by the question of how
could expectations of who was going to win the election change so completely (last-minute
swing)? Why people jumped on the bandwagon of the expected winner? To address these
questions, Noelle-Neumann (Elisabeth Noelle-Neumann, 1984, 1993) propose the
assumption of dual climate of opinion: firsthand observation of reality, and observation of
reality through the eye of the media.
The two sources of information are further conceptualized as dual climate of opinion:
the climate people observe directly from the reference groups, and the climate portrayed by
the media.
Individuals actively monitor the dual climate of opinion, and compute the distribution
of opinion, to avoid being punished by the society for holding the minority opinion. They will
fall silent if they feel their opinions are different from the dominating ideas of the mass media.
Further, the reinforcement of the process finally leads to the presence of spiral of silence.
Mass Media as the Third Level of Analysis of Spiral of Silence
There are at least three levels of the spiral of silence process: the individual level, the
reference group level, and the society level. Different level corresponds to its variables, for
example, the individual level corresponds to willingness to express, the fear of isolation and
the demographic variables; the reference group corresponds to the group size, the climate of
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the reference group, and the social capital of the reference group; the society level is
corresponding to characteristics of the culture and the patterns of the media, in fact, the
influence of the culture on the spiral of silence is thought to be the most fruitful area of new
research on the spiral of silence theory (Donsbach & Traugott, 2007; Scheufle & Moy, 2000).
However, it‟s necessary to distinguish individuals‟ media use behaviors and the
characteristics of the media, the individual‟s media use behavior is on the individual level,
while the characteristics of the mass media is on the third level.
One characteristics of the effects of the mass media (Donsbach & Traugott, 2007;
Gonzalez, 1988) is ubiquity effect, the communications are widespread in the respective of
space. The opinion expressed by the media comprises the climate of media which is
perceived by the individuals, and the individuals act according to the distribution of the
opinion expressed by media, so the content of the media is part of social structure, the
individuals‟ reactions to the media demonstrate that the social structure may influence the
individual behavior, just like the influence of culture.
Reference Group and Opinion Dynamics
Reference group was first used by Hyman (Hyman, 1942) to denote the group to which
individuals relate their attitudes to. It supplies references to the individuals (Shibutani, 1955).
Kelly (1952) emphasize two functions of reference group: first, other group members‟
behaviors supply standards for the judgment of individual‟s opinion; second, groups deliver
punishments or rewards to the group members.
Reference groups compete with mass media to influence individuals‟ opinions. The
influence of perceived dominance of an opinion is much bigger than that of the actual opinion.
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The study of Moy etc. demonstrate that reference group was a significant predictor of
willingness to express (Moy, Domke, & Stamm, 2001). Further, Krassa (1988) stated that a
main influence on willingness to express is the perceived distribution of opinion among the
social groups.
However, this is contrary to the proposition of pluralistic ignorance. Most scholars use
“pluralistic ignorance” to analyze the spiral of silence (Elisabeth Noelle-Neumann, 1993;
Taylor, 1982). It proposes that individuals fail to express because they falsely perceive their
own opinion to be in the minority. Apparently, the pluralistic ignorance will work only when
individuals lose supports of reference groups. That‟s to say, reference groups have been
controlled by the mass media.
Another two factors which compete with the mass media are „hard cores‟ and
„avant-gardes‟. Reference groups play an important role by providing a protective
environment for both the avant-gardes and the hard cores. Both „hard cores‟ and
„avant-gardes‟ resist the dominant opinion climate(Donsbach & Traugott, 2007; Elisabeth
Noelle-Neumann, 1984). Hard cores are those who stick to their minority opinion even as the
spiral of silence turns harsher, avant-gardes are special because they base their resistance on
ideological beliefs, for example, to promote new standpoints which are contrary with the
dominant opinion.
Reference group turns to be more important if the human behavior depends on the
information flow, not only the information flow within reference group, but also the
information flow inter-reference groups. However, most statistical models don‟t demonstrate
the dynamic influence of the interaction within the reference group and among different
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reference groups.
Research Framework
In the literature part, I review two main influential factors of individuals‟ willingness of
expressing. In the research framework part, I propose to analytically study it by adopting the
perspective of opinion threshold.
Opinion Threshold As the Tipping Point of Spiral of Silence
“Opinion threshold” is used to study the spiral of silence (Glynn & Park, 1997; Krassa,
1988), Krassa thinks that the behavioral threshold is according to the opinion threshold.
There‟s a threshold of individual‟s cognitive behavior. Schelling‟s papers (Schelling,
1971, 1972) about separation introduce the threshold into later studies, individuals have their
tipping points, when the proportion of different color residents living in the neighborhood
exceeds a threshold value, the person will move out to avoid being in the minority. Mark
Granovetter (Granovetter & Soong, 1983, 1986, 1988) follows Schelling‟s steps and proposes
threshold models, which assumes that individuals‟ behavior depends on the number of people
already engaging in that behavior. Threshold models were applied to explain the riot, the
residential segregation and spiral of silence, and the distribution of the opinion thresholds
determines the aggregate opinion (Granovetter & Soong, 1983, 1986, 1988).
There‟s also a phase transition of the public opinion: just like the ice becomes water
when the temperature is higher than the critical point (which is usually zero), the public
opinion is transformed from the liquid state (e.g. a morally-loaded question) to a solid state
(norms or dogmas) (Scheufle & Moy, 2000).
The crucial part of the study is to examine how individuals interact with the dual
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climate of opinions at the local scope gives rise to the emergence of spiral of silence. Two
competing factors (mass media and reference groups) will be analyzed.
The first one is the mass media as the third level factor. It will directly influence
individuals‟ perception about the mainstream opinions. The reference group as an aggregate
level variable is the second concern of this paper. will interact with the individual‟s choice: it
will punish the individuals if their opinions are different from the reference group, or else, it
will reinforce the individuals‟ behavior if their opinions are similar to the reference groups
(Kelley, 1952).
For those who hold the opinion of the mass media, the reference groups will reinforce
the influence of the mass media. On the contrary, for the people holding the opinion of
against that of the mass media will turn to the reference groups to find more supports, that is
to say, the reference group will reduce the influences of the mass media. It‟s straightforward
to integrating the thoughts above, propose and examine the first question of this paper:
RQ1: What‟s the boundary condition of a stable existence of spiral of silence?
Bottom-up interactions give rise to the global interaction. The scale of social
interaction both on the local scope and on the society level are constrained by the social
structure, especially the size of population, and the size of reference groups. It‟s reasonable to
identify how do size of population and reference groups influence the dynamics of spiral of
silence? We would propose that spiral of silence is size-dependent. Thus:
RQ2-1: How does the population size influence the emergence of spiral of silence?
RQ2-2: How does the size of reference groups influence the emergence of spiral of
silence?
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Although few studies pay attention to the dynamics of the number of people falling
silent over time, viewing the phenomenon as a process with the time dimension is very
important (Allport, 1937). Another concern of this paper is:
RQ3: How does the number of people falling silent change over time?
Method
Nonlinearity and the interplays between the aggregate level and the individual level are
the most important characters of the complex social phenomena, especially in the study of
spiral of silence. Just like Scheufele etc. (Donsbach & Traugott, 2007; Scheufle & Moy, 2000)
comment:
Future tests of the spiral of silence therefore need to include both aggregate-level
variables, such as cross-cultural comparisons, and individual-level predictors, such
as willingness to self-censor or fear of isolation. Most importantly, however, future
studies will have to examine the interactions between these aggregate-level
differences and the individual-level predictors of outspokenness.
Agent-based models (ABM) deal with these challenges well, it provides a theoretical
bridge between the micro level and the macro level. Different from the approach of
multi-level analysis, in ABM, agents directly interact with high level factors, which make the
logic quite straightforward, that is, individuals make reactions to the environment over time,
and the system evolve to emerge global changes. In the ABM of spiral of silence present in
this study, Agents act according to the dual opinion climate. The reference group and the
mass media, which are thought as the second level and the third level of the spiral of silence,
will influence the individuals directly.
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Agent-based Models
Agent-based modeling (Gilbert, 2008) is formally defined as a computational method
that enables the modelers to create, analyze, and experiment with models composed of agents
that interact within an environment. It (Macy & Willer, 2002) is designed to explore the
minimal conditions or assumptions required by specific social phenomenon which emerges at
a higher level of organization, which (Squazzoni, 2008) allows us to identify specific
mechanisms which can help outline the micro-macro links.
Agent-based model is beginning with Von Neumann‟s work (Neumann & Burks, 1966)
on self-reproducing automata, then it is widely used in physics, mathematics, biology, etc. Its
applications in social science are promoted by Conway‟s “game of life” model (Gardner,
1970), Schelling‟s model of neighborhood segregation (Schelling, 1971), and Axelrod‟s tit for
tat model (Axelrod & Hamilton, 1981). Agent-based models give rise to abundant research
focusing on opinion dynamics (Suo & Chen, 2008; Weisbuch, Deffuant, & Amblard, 2005;
Weisbuch, Deffuant, Amblard, & Nadal, 2002), innovation diffusion (Bullnheimer, Dawid, &
Zeller, 1998; Rosenkopf & Abrahamson, 1999; Strang & Macy, 2001), and it highlights the
domino and bandwagon effect, emergence, and criticality in the process.
The features of agent-based modeling (Gilbert, 2008) can be summed up by Nigel
Gilbert as the following: ontological correspondence, heterogeneous agents, representation of
the environment, agent interactions, bounded rationality, and learning. Agents interact with
little or no central authority or direction, but they follow simple rules, and they are
interdependent with each other, in addition to that, they are adaptive and backward-looking,
the agents adapt at both individual and population level (Macy & Willer, 2002).
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Two aspects of ABM should be highlighted. The first one is the attributes of the agent.
There are many colorful persons distributed in the squares, which are called agents. The
squares are named patches. Agents and patches have specific properties according to specific
models. The second one is the rule of evolution. “Rules are defined at the level of agents,
with the behavior of different agents being governed by its own set of rules. From these
agent-level behaviors, certain properties can emerge at the „macro-level‟ (Chen, Nagl, &
Clack, 2007)”.
Measure
It‟s very important to keep the model clear and concise, just as Robert Axelrod (1998)
denoted:
Although agent-based modeling employs simulation, it does not aim to
provide an accurate representation for a particular empirical application.
Instead, the goal of agent-based modeling is to enrich our understanding of
fundamental processes that may appear in a variety of applications.
There are many factors that will affect the spiral of silence, but this paper just focuses
on three variables according to three levels of analysis: the willingness to express, the
influence of the social network, and the influence of mass media.
The willingness to express
Every agent will begin with an variable named willingness to express. This variable
measure the people‟s choices of speaking out or not, but it‟s a continuous variable deprived
from a uniform normal distribution. The mean of the normal distribution is 0, and the
standard deviation of the distribution is 1.
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To define the willingness of express as normal distribution is reasonable, because most
of the statistical models used in the social science study assume that the social phenomenon is
distributed normally, except the Pareto distribution (which measure the power law
characteristic of the distribution, often used to measure the wealth‟s distribution in the
society).
I define the agents holding negative willingness to express as silent people; agents
carrying positive value of the variable are defined as the people who tend to speak out, and
their opinions are the same with the mainstream opinion. Willingness to express will change
over time, according to the changing climate of opinion, most people who find that the
climate opinion is hostile to their own opinions will fall silent to avoid being isolated (Glynn,
et al., 1997; E. Noelle-Neumann, 1974; Scheufle & Moy, 2000).
The influence of the reference groups
Every agent locates in its social network, and gets supports from the reference groups.
According to the idea of Noelle-Neumann (E. Noelle-Neumann, 1974), agents (individuals)
have a “quasi-statistical sense” which can accurately compute the opinion climate of its
reference group. In the model of this paper, each agent has a radius of observation, different
from the assumption which imagines the individual can monitor all the people in the society,
in my agent-based model, I assume that the agent only has bounded rationality, and it can
only monitor the climate of its reference group in the local scope.
To compute the influence of the reference groups, I sum the willingness to express of
each individual agent in its own radius of vision.
In the software Netlogo, using the code “in-radius vision”, the reference group of one
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agent can be classified as those whose distance from the caller is less than or equal to radius.
In my ABM of SOS, the radius of vision can be regulated, so that we can analyze the
dynamic influence of the size of reference groups.
The influence of the mass media
Because of the effects of the mass media (Donsbach & Traugott, 2007; Gonzalez, 1988)
involve consonance effect, cumulative effect, and ubiquity effect, mass media is just like the
culture, it‟s one important aspect of the society, so modelers do not need to establish specific
agents as mass media, on the contrary, the mass media is one kind of environment the agents
living in, so modelers can use the property of the patches to elaborate the influence of the
media, taking the media‟s influence are not totally equally distributed, each patch starts with
the variable named influence of media randomly taking from a dataset (0, 1, 2, 3, 4, 5), in the
dataset, 0 indicates there is no media‟s influence on the patch, from 1 to 5, the bigger the
number is, the bigger the influence of media on the patch is.
How does agent-based model work?
The agents act in respect with the local rules over time t, the number of agents is N, the
willingness to express is expressed by W, the nth agent‟s willingness to express at time t is
expressed as Wn,t, the nth agent stays at the nth patch, the influence of media at the nth patch
at time t is Mn,t, the nth agent‟s reference group‟s climate of opinion at time t is expressed as
Rn,t, and the coefficient of Mn,t is expressed as α, the coefficient of Rn,t is expressed as β, so
the the nth agent‟s willingness to express at time t is demonstrated in formula (1):
Wn , t = Wn , (t-1) + αMn , (t-1)+ βRn, (t-1)
(1)
I use NetLogo4.13 to model spiral of silence. Netlogo is a programmable modeling
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environment authored by Uri Wilensky(1999). It is suitable for modeling complex systems
developing over time to explore the connection between the micro-level behavior of
individuals and the macro-level patterns (Sklar, 2007; Wilensky & Rand, 2009).
Using Netlogo, I established an agent-based model of spiral of silence. Agent-based
model is also one kind of experiment, so it needs to control the variables to detect each
factor‟s influence. Given specific conditions, it‟s easy to test and justify the influence of the
reference group and the influence of the mass media respectively. At the initial state, there are
1000 agents randomly distribute in the patches, let its radius of vision be 3, so 3 as the radius
determines the boundary of each agent‟s reference group.
Results
The Boundary Conditions of Spiral of Silence
The first research question concerns about the boundary conditions of spiral of silence.
According to literature review, reference groups compete with mass media to influence
individuals‟ opinion dynamics. The ratio of the influence of reference groups and that of the
mass media, supplies a good way to identify the boundary condition I am aiming to look for.
Following this line of thought, I define the reference groups as strong reference groups when
α/β<1, and moderate reference groups when α/β=1, and weak reference groups when α/β>1.
To address the first research question, I set radius=3, N=1000, α/β equals 0.1, 1, and 10,
to explore the result. The primary results are showed in Figure 1-3.
[Insert Figure 1 here]
Model 1 with α/β= 0.1, 1, and 10, are all run five times. As Figure 1 shows, in general,
the results are mixed. To be specific, when α/β=0.1 (strong reference groups, weak mass
media), the results are unstable. To more fully explore this pattern, I establish Model 2 by
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fixing α/β= 0.1, radius=3 and N=1000. The results of Model 2 are demonstrated in Figure 2,
which confirm our judgment about the result of strong reference groups. So given the
boundary condition of strong reference groups (compared with the influence of mass media),
there is no guarantee for the stable existence of spiral of silence.
[Insert Figure 2 here]
To further test the influence of moderate reference groups and weak reference groups, I
establish model 3 by setting α/β=1, 10; radius=3; N=1000. The results of Model 3 are showed
in Figure 3. It‟s obvious that when the influence of reference groups is moderate or weak, the
global regularity of spiral of silence stably emerge from individual behaviors. Further, the
population with only moderate reference groups falls silent faster than the population with
weak reference groups. Which implies that when the influence of reference groups are too
strong (see Model2 and Figure 2), the population may fall silent. Yet population with strong
mass media, compared with reference groups, the stronger the influence of reference groups
is, the faster the population fall silent.
[Insert Figure 3 here]
Individual‟s interactions with the reference groups at the local scope will influence the
spiral of silence, whether reference group may reinforce or diminish the spiral of silence
depending on the ratio of the mass media and that of the reference groups.
The influence of mass media
[Insert Figure 4 here]
Given the population size=1000, the vision=3, beta=0, alpha=0.02. The dynamical
process is illustrated in Figure 4. The figure 4 plots the number of the silent people on the
vertical Y-axis against time on the horizontal X-axis.
It takes 37 ticks for all the people falling silent. Further, the cumulative curve of the
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number of people falling silent over time can be fitted with exponential function. The
influence of mass media is very straightforward. It just makes more and more people falling
silent.
The influence of reference group
Given the population size=1000, vision=3, beta=0.02, alpha=0, I find the result is
complex and unstable. Different from the mass media, the influence of the reference group is
much complicated. The reference groups have two opposing functions: to punish or to reward
the individuals, both of which make the agents centered in the cluster reinforce their own
opinions. The reference group promotes the agents to convergence, and the larger the size of
the reference group is, the sooner the model will end.
Without the influence of the mass media, the convergence of opinions is based on
agents‟ geographical distribution. The agents who are close with each other will be affected
first. After steps of contagion, the silent people and the talking people are separated clearly
into two clusters. The opposing two parts combat with each other across boundaries, and the
strong side will dominate the opinion by surrounding and isolating the others.
[Insert Figure 5 here]
Figure 5 demonstrates the separation and convergence. Rather than an emergence of
spiral of silence, “spiral of speaking-out” happened. In this process, the geographical
distribution really matters. The talking people succeed by surrounding the other side. It‟s
crucial to put the right agent in the right place, which is like the game of chess (played with
181 black pieces and 180 white pieces). Individuals can‟t scan the global distribution of
public opinion. They can only monitor the local climate of public opinion in his vision. The
evolution of spiral of silence driven only by the reference group is path-dependent. It relies
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on the structure of the social networks, that is, the geographical distribution of the agents.
The Size of Reference Groups
To test the robustness of this pattern, I adjust the vision of the agents as 2, 4, 6. When
the vision is 2, the agents only take account of the opinion of the agents who are in the radius
of 2 patches. The larger vision is, the larger the size of the reference groups is, the more
information agents can get from the neighborhood.
[Insert Figure 6 here]
The results indicate that, with the other conditions fixed, when the size of the
reference groups is bigger, the time used by the model reduces (see Figure 6). This is
reasonable, when the reference group is big enough, it takes less time for the individual to
find the dominant opinion and get on the bandwagon. The larger size of the reference groups
is, the faster the population reach consensus.
The Size of Population
To test the influence of population size, I set the population of the agents as 1000, 1500,
2000. The larger the population size is, the larger the density of the reference groups is, the
more information agents get from the neighborhood.
Figure 7 indicates that the larger the population size is, the sooner all the people falling
silent.
[Insert Figure 7 here]
The Evolution of Opinion Dynamics over Time
Figure 7 demonstrates how the number of people falling silent decreases over time. At
first, the number of people falling silent grows fast, but the growth rate gets slower and
slower over time.
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[Insert Figure 8 here]
Further test show the numbers of people falling silent each time are distributed as
power law. Most of the time, there are very few people falling silent, but sometimes, a large
number of people fall silent together. The number of people falling silent is scale-free.
[Insert Figure 9 here]
Using the simulation data, Figure 8 plots the percentage of the number of the people
falling silent each time and the number of people falling silent each time. I fit the data using
Pareto function
β
to test the distribution of the number of people falling
silent per time, the statistic parameter are list as the following table:
[Insert Table 1 here]
The shows that the parameters are all significant (p < 0.001), the data can be fitted
very well by the Pareto distribution function, using x to express the number of people falling
silent each time, the probability dense function y is：
So it‟s reasonable to draw a conclusion that the number of the people who fall silent per
time is Pareto distribution, which is highly skewed. The bigger the size is, the smaller its
probability is, just like the wealth distribution in the society.
Conclusion and Discussion
With the perspective that an emergent behavior or emergent property can appear when
actors interact in an environment, this study attempted to interpret how individuals‟
interaction with the dual climate of public opinion (e.g. the mass media and the reference
groups) in the local level can lead to the global phenomenon of spiral of silence.
Using the agent-based modeling method, this study provides a theoretical test of spiral
of silence theory: the spiral of silence as a global macroscopic regularity of the society is
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generated from the heterogeneous individual behaviors.
The process is bottom up, individuals act according to the local principle (the capability
of the agents is local, no agent has global information), and fundamental social structures and
group behaviors emerge from the interactions of individual agents. This process (Epstein &
Axtell, 1996) is usually called emergence. There is no doubt that the spiral of silence is an
emergence of collective falling silent from the agents‟ individual behavior in response to
social influences or selection pressure.
Besides, there are several interesting patterns emerging from the study:
First, individuals‟ interactions with the opinion of the mass media at the local scope
give rise to the spiral of silence. According to the findings of this study, the effect of the mass
media is very obvious and stable, which support the idea that public opinion influenced by
the mass media can be defined as a kind of social control (Scheufle & Moy, 2000).
Second, the effect of the reference group is complex but enlightening. The findings
indicate that the agents who are close with each other will converge, and accompanied by the
convergence, the agents will quickly be separated into two opposing parts with regard to the
opposite opinions. Individual‟s interaction with the reference groups at the local scope may
reinforce the mass media‟s effect on the spiral of silence, but it can also countervail or even
reverse the spiral of silence driven by the mass media. This finding is corresponding to the
primary function of the reference groups: it will punish the individuals if their opinions are
different from the reference group, or else, it will reinforce the individuals‟ behavior if their
opinions are similar to the reference groups (Kelley, 1952). Reference group promotes the
agents to convergence, and the larger the size of the reference group is, the sooner the model
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20
will end. The bigger the density of reference groups is, the sooner the spiral of silence will
emerge.
Third, the dynamics of spiral of silence is a curve of polynomial function, which is
steeper than the exponential curve. It‟s relevant to the growth rate of the number of people
falling silent per time. The reinforcement driven by the dual climate of public opinion will
diminish over time (Taylor, 1982). Further, the number of people falling silent per time is
distributed as power law.
The contribution of this study falls in the area of analytically examining the boundary
conditions and exploring the characteristics of the spiral of silence. The main limitation of
this research is that I use the random network as an environment to test the spiral of silence,
while the real social network maybe small-world network (Watts & Strogatz, 1998) or
scale-free network (Barabasi & Crandall, 2003), the difference of the social network may
influence the research conclusion, especially the conclusion about the role of reference group.
This will be done in my future work.
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Number of people
falling silent
Running head: EMERGENCE OF SPIRAL OF SILENCE
1000
800
600
400
200
0
1 3 5 7 9 1113151719212325272931333537
Time
Figure 1
25
α/β=0.1
α/β=0.1
α/β=0.1
α/β=0.1
α/β=0.1
α/β=1
α/β=1
α/β=1
α/β=1
α/β=1
α/β=10
α/β=10
α/β=10
Reference groups, mass media, and spiral of silence (α/β= 0.1, 1, and 10;
1000
900
800
700
600
500
400
300
200
100
0
1
13
25
37
49
61
73
85
97
109
121
133
145
157
169
181
193
205
217
229
241
253
265
277
Number of people falling silent
radius=3; N=1000)
Time
Number of people falling silent
Figure 2 Strong Reference Groups, Weak Mass Media (α/β=0.1; radius=3; N=1000)
1000
900
800
700
600
500
400
300
200
100
0
/=10
/=1
1
3
5
7
9
11 13 15 17 19 21 23 25 27
Time
Running head: EMERGENCE OF SPIRAL OF SILENCE
26
Number of people falling
silent
Figure 3 Global Regularity and Strong Mass Media (α/β=1, 10; radius=3; N=1000)
1200
1000
800
600
400
200
0
1 3 5 7 9 1113151719212325272931333537
Time
Figure 4
the model of mass media (α= 0.02, β= 0, 10; radius= 3; N= 1000)
Figure 5
Convergence of close agents
Number of people falling silent
Running head: EMERGENCE OF SPIRAL OF SILENCE
27
1000
900
800
700
600
500
400
Radius=6
Radius=4
300
Radius=2
200
100
0
1 4 7 101316192225283134374043464952555861646770737679
Time
Figure 6 Size of Reference Groups and Time of Convergence (==0.002, population=1000,
radius= 2, 4, 6)
Number of people falling silent
2500
Population=1000
2000
1500
Population=2000
1000
500
Population=1500
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
Time
Figure 7 Population Size and Spiral of silence (==0.02, population=1000, 1500, 2000,
radius= 3)
Running head: EMERGENCE OF SPIRAL OF SILENCE
Figure 8
Figure 9
28
Number of people falling silent per time
Pareto distribution of Number of the people falling silent each time
Table 1
Nonlinear regression analysis of the distribution of the number of the people falling silent
each time
Parameter Estimate
R Squared
Standard Error
t Statistic
P-Value
182.52
4.13
44.23
1.1×10-7
2.28
0.124
18.37
8.8×10-6
0.998
Running head: EMERGENCE OF SPIRAL OF SILENCE
29