=Paper=
{{Paper
|id=Vol-2792/paper13
|storemode=property
|title=Opinion Dynamics Models with Noise
|pdfUrl=https://ceur-ws.org/Vol-2792/paper13.pdf
|volume=Vol-2792
|authors=E. Konovalchikova,Yu. Dorofeeva
}}
==Opinion Dynamics Models with Noise==
https://ceur-ws.org/Vol-2792/paper13.pdf
Opinion Dynamics Models with Noise
E. Konovalchikova1 and Yu. Dorofeeva2
1
Karelian Research Centre RAS, Laboratory for Digital Technologies
in Regional Development, Department for Multidisciplinary Scientific Research,
11 Pushkinskaya St., 185910 Petrozavodsk, Russia
2
Petrozavodsk State University, 33 Lenin Ave., 185910 Petrozavodsk, Russia
Abstract. The paper analyses noise effect on opinion dynamics be-
haviour in connection with different noise distribution parameters. The
coefficient of variation, which determines the degree of noise homoge-
neity, was chosen as the principal characteristic of noise. In this study,
a dependence was detected between the degree of noise homogeneity
and the consensus-reaching time in Friedkin-Johnsen and Hegselmann-
Krause opinion dynamics models.
Keywords: Opinion dynamics · Consensus · Noise · Modelling · Heg-
selmann-Krause Dynamics · Friedkin-Johnsen Dynamics · Coefficient of
Variation · Stochastic Process
1 Introduction
Individual opinion dynamics in social systems is a complex process, which shapes
the relationships among individuals inside the system depending on similarities
and differences between individuals, their inner conflicts, presence or absence
of leaders in groups. Opinion dynamics is construed as the process of opinion
formation in a group through a fusion of individual opinions in which interacting
agents within a group continuously update and fuse their opinions on the same
issue based on the established fusion rules and reach a consensus, polarization, or
fragmentation in the final stage [10]. Opinion formation in a group is a stochastic
process, since agents’ opinions are random variables with some distribution.
In a majority of mathematical models for opinion dynamics in social systems
[2,3,8,19,26,27], the final state of the dynamics is either a perfect consensus or
a division of the population into groups whose members share a certain opin-
ion [22]. In reality however, public opinion in social groups does not reach such
ideal states of complete consensus [30]. To obtain more realistic results from the
modelling of opinion dynamics in social groups, an additional random element
is introduced, termed noise impact or, simply, noise. The authors of [28] view
noise as “free will” owing to which individuals can change their opinion ran-
domly, while in [39] noise is interpreted as a replacement of individuals with new
ones in groups of non-fixed size. Noise can also be interpreted as partial loss
of information or its distortion in the course of interactions for various reasons.
Research of the noisy opinion dynamics models is a way to analyse the noise
Copyright © 2020 for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0).
resistance of opinion dynamics and to expose new, previously not investigated,
complex phenomena in social systems generated by various noise interventions.
This is particularly important when studying how opinions are formed in vari-
ous societies in the situation where many activities are migrating to the digital
space, where there are multiple factors that influence, directly or indirectly, the
process of decision making by individuals or groups.
This article explores the stochastic process of opinion formation in the pres-
ence of noise using the Friedkin-Johnsen and Hegselmann-Krause opinion dy-
namics models as examples. The two models were chosen owing to their contin-
uous nature, availability of an individual parameter for agents in groups, which
represents the agent’s susceptibility to influence from the rest of the group in
the former dynamics, and the confidence threshold in the latter. The coefficient
of variation, which defines the degree of noise homogeneity, was chosen as the
principal characteristic of noise. The study aims to reveal the relationship be-
tween the degree of noise homogeneity and the consensus-reaching time in the
cases of uniform and normal noise distribution.
The article is structured as follows. The second section briefly reviews the
literature on the topic. The third section examines the effect of noise on the
convergence of Friedkin-Johnsen and Hegselmann-Krause opinion dynamics for
the cases of uniform and normal noise distribution taking into account changes
in the coefficient of variation. The conclusions summarise the key findings of the
study.
2 Literature Review on the Subject
The papers [10,22,34] give a classification of the existing models for opinion
dynamics based on the format of opinion expression by agents, according to
which the pool of opinion dynamics models can be divided into continuous or
discrete models. The authors of [10,34] additionally distinguish a group of hybrid
models. The group of continuous models includes the classical DeGroot model
[1,8], the Deffuant et al. model [9], the Hegselmann-Krause model [19], as well
as the Friedkin-Johnsen model [15,16], which is one of the few to have been
experimentally validated in small and medium-size groups [17]. A distinctive
feature of continuous models for opinion dynamics is that the opinion of any
agent is expressed as a continuous variable, so that situations can be simulated
where the opinions of agents in a group regarding an issue vary continuously over
time, taking any values ranging from “fully agree” to ”completely against” [28].
Discrete opinion dynamics models, studied in the papers [6,7,14,20,31,35,36,38],
are used to model situations with a limited choice of solutions for an issue. The
group of hybrid models encompasses the continuous opinions and discrete actions
(CODA) model [23,24,25], the DeGroot-Friedkin model [21], and the model for
public opinion dynamics in an online-offline social network context [11,13].
Exposure to noise in the process of opinion formation in social systems plays
a key role, wherefore one of research alleys is the design and analysis of opin-
ion dynamics models with noise [4,5,12,28,41]. The effects of noise on opinion
formation are the most commonly studied by the continuous Deffuant [4,28,29]
and Hegselmann-Krause [30,37,41] models. Both models employ the bounded
confidence concept, which implies that agents can influence one another only if
the distance between their opinions is below a certain threshold, and the mecha-
nism for common opinion formation based on the averaging rule. The fundamen-
tal difference between these models is the mode of interactions between agents
in groups. In the Hegselmann-Krause model, agents interact in a large group,
whereas the Deffuant model focuses on pairwise interactions only. The original
Deffuant and Hegselmann-Krause models are studied and their similarities and
distinctions are described in the papers [22,40].
Fig. 1. The original Friedkin-Johnsen opinion dynamics.
The paper [30] explores the noisy Hegselmann-Krause model for opinion
dynamics where agents are allowed, with certain probability, to spontaneously
change their opinion to another one, selected randomly inside the opinion space.
This paper analysed how the opinion dynamics evolved in the context of un-
bounded and bounded random opinion jumps inside the whole opinion space or
in a limited interval centred around the current opinion. The authors of [30] have
also scrutinised the similarities and differences of this noisy Hegselmann-Krause
opinion dynamics and the matching Deffuant opinion dynamics, described pre-
viously in [4,29].
The convergence conditions for the noisy Hegselmann-Krause opinion dy-
namics are the main subject for the papers [37,41]. The paper [37] provides a
rigorous theoretical analysis of the consensus behaviour of opinion dynamics in
noisy environments and demonstrates that noise helps to “synchronise” opinions.
The main difference of this article from the above works on the study of the
behavior of the dynamics of opinions with noise is a new approach to the study
of the influence of noise, based on the dependence of the rate of convergence of
the dynamics on the degree of homogeneity of the noise. We explore the effect
of homogeneous and heterogeneous noise on the behaviour of known opinion
dynamics provided that the noise is a random variable with uniform or normal
distribution.
3 Continuous Opinion Dynamics Models with Noise
In this section we will examine two continuous opinion dynamics models which
use the averaging rule in opinion formation, and investigate the effect of noise
perturbations represented by independently and identically distributed random
variables on the convergence of these dynamics. Firstly we take the Friedkin-
Johnsen opinion dynamics for the model, and then we analyse the Hegselmann-
Krause model, whose main distinction from the former is that it uses the bounded
confidence concept. The convergence of both opinion dynamics will be investi-
Fig. 2. Noisy Friedkin-Johnsen dynamics with CV = 0.2.
gated from the point of view of the effect produced on the convergence of agents’
opinion dynamics by the noise variation coefficient, which defines the degree of
noise homogeneity.
3.1 Friedkin-Johnsen Model with Noise
The Friedkin-Johnsen model describes the stochastic process of opinion forma-
tion in a social group with n agents under the condition that each agent has
an opinion [18] of their own on the issue, which tends to change in the process
of discussions depending on the agent’s susceptibility to the opinions of other
members of the group.
In the original model, the opinion at current time instant is formed as the
mean weighted value of the agent’s initial opinion and the opinions of all agents
at the preceding time instant according to the formula [15,16]:
y(k + 1) = ΛW y(k) + (I − Λ)y(0), (1)
where yi (k) ∈ [0, 1] — is the value of i-th agent’s opinion at k-th time in-
stant; y(0) = (y1 (0), y2 (0), . . . yn (0))T is the vector of the agents’ initial opinions;
y(k) = (y1 (k), y2 (k), . . . yn (k))T is the vector of the agents’ opinions at k-th time
instant; W = [wij ] is the stochastic matrix of the agents’ influences on one an-
Pn
other (0 ≤ wij ≤ 1, wij = 1, i, j = 1, . . . , n); Λ = I − diag(W ) is the diagonal
j=1
matrix of the agents’ susceptibility to the influence (opinion) of others.
In the original opinion dynamics in a group of n = 100 agents holding differ-
ent opinions xi ∈ [0, 1] it is already since the third stage of the discussion that
all agents in the group hold the same opinions, i.e. reach a consensus [18] (see
Figure 1).
Fig. 3. Noisy Friedkin-Johnsen dynamics with CV = 0.33.
When noise is added to the model for the dynamics (1), the values of agents’
opinions may go beyond the interval [0, 1]. To exclude such situations, a boundary
condition for absorption is used, which stipulates that all opinion values beyond
the interval [0, 1] are equated to 0 if the values are to the left of 0, or to 1 if
the values are to the right of 1. Thus, agents’ opinion dynamics in the noisy
Friedkin-Johnsen model is given by the following formula:
0, yi∗ (k + 1) < 0,
yi (k + 1) = yi∗ (k + 1), 0 ≤ yi∗ (k + 1) ≤ 1, (2)
1, y ∗ (k + 1) > 1,
for any i ∈ {1, 2, . . . , n} and k ≥ 0, where
yi∗ (k + 1) = ΛW yi (k) + (I − Λ)yi (0) + ξi (k) (3)
for any i ∈ {1, 2, . . . , n} and k > 0. The noises {ξi (k)} are independently and
identically distributed random variables with the distribution function F .
Fig. 4. Noisy Friedkin-Johnsen dynamics with CV = 0.7.
Consider the effect of noise homogeneity on the convergence of the dynamics
(2)-(3) in the case of a uniform and normal noise distribution. Noise is said to
be homogeneous if its coefficient of variation is below 0.33 [32,33]. In the case of
uniform noise distribution, the opinion dynamics (2)–(3) would always converge
if noises are homogeneous, but the convergence time would increase compared to
the original dynamics (1). Where noises are heterogeneous, the opinion dynamics
(2)–(3) converges to a consensus if the coefficient of variation takes a value in
the interval [0.33, 0.65), and diverges if the coefficients of variation satisfy the
condition CV ≥ 0.65. Numerical simulation results indicate that in the case of
reaching a consensus it is not only the time of convergence that is enlarged by an
increase in the coefficient of variation but also the scatter of opinions. The con-
vergence behaviour of the opinion dynamics (2)–(3) in a group of n = 100 agents
given different coefficients of variation is shown in Figures 2, 3 and 4. The scatter
of opinions in Figure 4 can be interpreted as a chaos of opinions inside the group.
Table 1 shows the numerical simulation results for the convergence period tU of
the Friedkin-Johnsen dynamics for different coefficients of variation in the case
of uniform noise distribution. In the normal noise distribution case, numerical
simulation yields similar results: the opinion dynamics (2)–(3) converges to a
consensus when the coefficient of variation takes values that satisfy the inequal-
ity CV < 0.5, and does not reach consensus if CV ≥ 0.5. Observe that if noise
in the opinion dynamics (2)–(3) has a normal distribution, a chaos of opinions
Table 1. Friedkin-Johnsen dynamics convergence period in the case of uniform noise
distribution.
CV 0.01 0.15 0.20 0.3 0.33 0.55 [0.65, 1]
tU [6; 100] [12; 100] [20; 100] [21; 100] [30; 100] [85, 100] —
would occur sooner than if noise is distributed uniformly. Another distinction of
the uniform distribution case is the discontinuous nature of consensus, i.e. there
exist several convergence regions (see Figure 5). As the coefficient of variation
Table 2. Friedkin-Johnsen dynamics convergence period in the case of normal noise
distribution.
CV tN
0.01 [3; 45] ∪ [47; 100]
0.15 [5; 45] ∪ [50; 100]
0.20 [5; 25] ∪ [37; 100]
0.30 [5; 30] ∪ [45; 85] ∪ [90; 100]
0.33 [3; 10] ∪ [13; 20] ∪ [21; 25] ∪ [35; 54] ∪ [56; 65] ∪ [68; 72] ∪ [75; 80] ∪ [85; 100]
[0.5, 1] —
grows in the opinion dynamics (2)–(3), the number of such convergence regions
increases, as shown in Table 2.
Thus, numerical simulation of the noise variation coefficient effect on the
convergence of the opinion dynamics (2)–(3) revealed that irrespective of the
noise distribution pattern, the dynamics converges to a consensus when noise is
homogeneous, and either converges or diverges in the presence of heterogeneous
noise depending on the variation coefficient. Where noise has a normal distribu-
tion, the convergence pattern of the dynamics is discontinuous, and a chaos of
opinions would occur sooner than in the case of uniform noise distribution.
3.2 Noisy Hegselmann-Krause Model
The Hegselmann-Krause opinion dynamics model is based on the bounded con-
fidence mechanism, according to which two agents influence each other only if
the distance between their opinions is below a certain threshold [30]. Keeping
this concept in mind, the stochastic process of i th agent’s opinion formation at
any given time instant is described by the mean weighted value of i th agent’s
opinion and the opinions of other agents belonging to that agent’s circle of trust,
−1
X
yi (k + 1) = |N (i, y(k))| yj (k), i ∈ {1, 2, . . . , n} , (4)
j∈N (i,y(k))
where yi (k) ∈ [0, 1] i th agent’s opinion at k th time instant and
N (i, y(k)) = {1 ≤ j ≤ n : |yj (k) − yi (k)| ≤ ε}
Fig. 5. Normal distribution of noise in the Friedkin-Johnsen dynamics with CV = 0.33.
is the set of agents belonging to i th circle of trust with the radius ε [19]. Figures 6
and 7 ı̈illustrate the behaviour of the opinion dynamics (4) in a group of 100
agents for different values of the radius of trust ε.
The Hegselmann-Krause dynamics convergence depends on the value of the
confidence threshold ε.
Fig. 6. The original Hegselmann-Krause dynamics with ε = 0.05.
A low level of trust among agents in the group leads to opinion fragmenta-
tion, i.e. the group splits into several subgroups holding similar opinions, inside
which the participants reach a consensus, whereas with a high confidence thresh-
old a consensus would be reached inside the original group. Thus, the original
Hegselmann-Krause opinion dynamics would converge to a consensus if ε > 0.5
and would diverge if ε ≤ 0.5. When ε values are low, the following pattern is ob-
served: as the confidence threshold declines, the speed of opinion fragmentation
in the group and the number of subgroups holding different opinions increase.
To describe the noisy stochastic process of opinion formation, noise ξi (k),
represented by a random variable with a distribution function F , is added to the
original model (4) for the formation of i th agent’s current opinion.
Fig. 7. The original Hegselmann-Krause dynamics with ε = 0.6.
In this case i th agent’s opinion dynamics in the noisy Hegselmann-Krause
model has the form
0, yi∗ (k + 1) < 0,
yi (k + 1) = yi∗ (k + 1), 0 ≤ yi∗ (k + 1) ≤ 1, (5)
1, y ∗ (k + 1) > 1,
for any i ∈ {1, 2, . . . , n} and k ≥ 0, where
−1
X
yi∗ (k + 1) = |N (i, y(k))| yj (k) + ξi (k), i ∈ {1, 2, . . . , n} (6)
j∈N (i,y(k))
with noise {ξi (k)} for any i ∈ {1, 2, . . . , n} and k > 0 [42].
The analysis of the variation coefficient effect on the convergence rate of
the opinion dynamics (5)–(6) in the case of uniform and normal noise distribu-
tion {ξi (k)} showed that the effect of the confidence threshold ε on the opinion
fragmentation process remains as in the original model. An increase in the co-
efficient of variation extends the time it takes to reach a consensus in the initial
group or in the subgroups formed through fragmentation. Numerical simulation
results for the convergence time of the opinion dynamics (5)–(6) in the uniform
Table 3. Hegselmann-Krause dynamics convergence period in the case of uniform noise
distribution.
CV ε = 0.05 ε = 0.2 ε = 0.5 ε = 0.6 ε = 0.9
0.01 TU = 8 TU = 11 TU = 9 TU = 8 TU = 2
0.20 TU = 15 TU = 18 TU = 22 TU = 13 TU = 5
0.33 TU = 60 TU = 28 TU = 25 TU = 27 TU = 20
0.60 TU = ∞ TU = 75 TU = 88 TU = 80 TU = 64
0.80 TU = ∞ TU = ∞ TU = 97 TU = 84 TU = 65
0.90 TU = ∞ TU = ∞ TU = 160 TU = 150 TU = 145
1 TU = ∞ TU = ∞ TU = ∞ TU = ∞ TU = 320
noise distribution case are shown in Table 3. Thus, in the case of uniform noise
distribution the opinion dynamics (5)–(6) converges to a consensus if noise is
homogeneous: for the confidence threshold values ε < 0.5 convergence is possi-
ble only inside the subgroups formed through fragmentation, and for ε ≥ 0.5 it is
possible inside the whole group. Where noise is heterogeneous, the vector of the
Hegselmann-Krause dynamics depends on the value of the confidence threshold
ε: the dynamics diverges if ε is low and converges if its values are close to 1.
Fig. 8. Hegselmann-Krause dynamics for ε = 0.05 and CV = 0.2.
Similar results were obtained in the numerical simulation of the opinion dy-
namics (5)–(6) for the case of normal noise distribution. According to the results
in Table 4, in the presence of homogeneous noise the opinion dynamics reaches
a consensus either in some subgroups formed through fragmentation or in the
whole group depending on the value of the confidence threshold ε.
With a high confidence threshold, near 1, the opinion dynamics (5)–(6) rea-
ches a consensus in the initial group irrespective of the degree of noise homo-
geneity, but the convergence period is significantly extended by an increase in
the coefficient of variation.
Table 4. Hegselmann-Krause dynamics convergence period in the case of normal noise
distribution.
CV ε = 0.05 ε = 0.2 ε = 0.5 ε = 0.6 ε = 0.9
0.01 TN = 10 TN = 13 TN = 10 TN = 2 TN = 2
0.2 TN = 30 TN = 25 TN = 25 TN = 15 TN = 5
0.33 TN = 70 TN = 52 TN = 50 TN = 47 TN = 30
0.6 TN = ∞ TN = ∞ TN = 90 TN = 85 TN = 67
0.8 TN = ∞ TN = ∞ TN = 105 TN = 90 TN = 70
0.9 TN = ∞ TN = ∞ TN = 165 TN = 160 TN = 150
1 TN = ∞ TN = ∞ TN = ∞ TN = ∞ TN = 350
The behaviour of the Hegselmann-Krause dynamics with homogeneous and
non-homogeneous noise, where the confidence threshold in a group of 100 agents
is 0.05, is shown in Figures 8 and 9. They demonstrate that an increase in the
coefficient of variation is accompanied by a decrease in the number of subgroups
formed through fragmentation and by an increase in the scatter of opinions at
the stages preceding consensus inside each of the groups. A similar situation is
observed with any confidence threshold values that satisfy the condition ε < 0.5.
Fig. 9. Noisy Hegselmann-Krause dynamics for ε = 0.05 and CV = 0.33.
Thus, irrespective of the noise distribution pattern, an increase in the de-
gree of noise homogeneity leads to an extension of the consensus-reaching time
in the opinion dynamics (5)–(6), while the effect of the confidence threshold ε
on consensus attainment in the noisy Hegselmann-Krause dynamics remains as
in the original model. Numerical simulation of the effect of the noise variation
coefficient on the convergence of the dynamics (5)–(6) revealed that where the
confidence threshold is high, near 1, the opinion dynamics would reach a con-
sensus irrespective of the variation coefficient, and the latter influences only the
speed of approaching consensus.
4 Conclusions
This paper explored the effect of noise on the convergence rate of the Friedkin-
Johnsen and Hegselmann-Krause opinion dynamics, chosen owing to their con-
tinuous nature, availability of an individual parameter for agents in groups, which
represents the agent’s susceptibility to influence from the rest of the group in
the former model of dynamics, and the confidence threshold in the latter. It
was found through this study that irrespective of the noise distribution pattern
the time it takes to reach a consensus is extended by an increase in the noise
variation coefficient in both opinion dynamics. In other words, as noise becomes
more heterogeneous, it gets more difficult for agents in a group to form a common
opinion on an issue.
Where noise is homogeneous, the Friedkin-Johnsen opinion dynamics always
reaches a consensus, while with non-homogeneous noise the dynamics would ei-
ther converge or diverge depending on the coefficient of variation. Where noise
has a normal distribution, this dynamics would converge in a discontinuous man-
ner, and the convergence discontinuity periods can be interpreted as “temporary
conflicts” inside the group in the process of common opinion formation.
Studying the noisy Hegselmann-Krause opinion dynamics we found that the
agents’ confidence threshold had the same value as in the original model. When
the confidence threshold is 0.5, consensus can be reached inside the initial group,
otherwise a fragmentation of opinions takes place, i.e. the group of agents splits
into several opinion-sharing subgroups.
Numerical simulations with homogeneous noise showed that depending on the
confidence threshold level the dynamics would converge either in the subgroups
formed through fragmentation or in the whole group, but the time needed to
reach a consensus would increase as noise gets less homogeneous. In the case of
heterogeneous noise, opinion dynamics in a group would reach a state of “chaos”
if confidence thresholds are low, whereas high confidence thresholds would lead
the dynamics to a consensus, but the convergence would be slower. Thus, at
any level of noise homogeneity members of consolidated groups, which have a
high confidence threshold, would anyway reach a consensus in dealing with an
issue, whereas a divided group would be pushed towards ”chaos” even by “slight”
noise.
Numerical simulation of the effect of noise on the behaviour of both dynamics
revealed that the period of the dynamics’ convergence to a consensus was greater
in the case of normal noise distribution compared to uniform distribution. The
difference in the dynamics convergence time suggests that noise with a normal
distribution has a “stronger” effect on the dynamics’ behaviour than noise with
a uniform distribution.
It is noteworthy that in the presence of noise consensus would be more fre-
quently attained in the Friedkin-Johnsen dynamics model than in the Hegsel-
mann-Krause model, due to the mode of interactions among agents inside the
group. In the former mmodel of dynamics an interaction involves all agents in
the group, whereas in the latter it is only among agents belonging to the circle
of trust. Thus, a team split into several subgroups is more susceptible to noise
effects than the whole team together. Furthermore, the more subgroups there
are in a team the more susceptible it is to the influence of noise. This pattern
demonstrates that the time needed to reach a consensus is directly dependent
not only on noise homogeneity but also on the team’s “unity”
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