VOL. E105-B NO. 8
AUGUST 2022
The usage of this PDF file must comply with the IEICE Provisions
on Copyright.
The author(s) can distribute this PDF file for research andeducational (nonprofit) purposes only.
Distribution by anyone other than the author(s) is prohibited.

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

990

PAPER

Modeling Polarization Caused by Empathetic and Repulsive
Reaction in Online Social Network
Naoki HIRAKURA†a) , Student Member, Masaki AIDA†b) , and Konosuke KAWASHIMA†c) , Fellows

SUMMARY While social media is now used by many people and plays
a role in distributing information, it has recently created an unexpected
problem: the actual shrinkage of information sources. This is mainly due tothe ease of connecting people with similar opinions and the recommendation
system. Biased information distribution promotes polarization that divides
people into multiple groups with opposing views. Also, people may receive
only the seemingly positive information that they prefer, or may trigger
them into holding onto their opinions more strongly when they encounter
opposing views. This, combined with the characteristics of social media,
is accelerating the polarization of opinions and eventually social division.
In this paper, we propose a model of opinion formation on social media tosimulate polarization. While based on the idea that opinion neutrality is only
relative, this model provides new techniques for dealing with polarization.
key words: social media, polarization, filter bubble

1.

Introduction

Today, a lot of social media streams exist and play importantroles in information distribution. Unlike traditional mass
media, social media allows users to collect information oninteresting topics more efficiently. However, the increased
convenience has been accompanied by the unexpected problem of the diminution of actual information sources. The
main reasons for this are users’ freedom to choose which
streams to follow and the recommendation system, which
delivers information thought to be of interest to individual
users. Polarization is a concern as only users with similarideas tend to communicate with each other, which creates abiased information distribution environment [1]. Polarization is the emergence of groups with conflicting opinions on
a topic. As polarization strengthens, it leads to an intensification of slander and defamation between the conflicting
groups. Polarization was confirmed on several topics. For
example, polarization on three major topics: gun control,same-sex marriage, and climate change [2] and secular andIslamist groups [3].
Networks play a significant role in the formation of
the individual’s opinion, and many models have been proposed [4], [5]. In this paper, we propose a model of opinion
formation on social media to simulate polarization. Thismodel incorporates users’ empathetic and repulsive reacManuscript received September 12, 2021.
Manuscript revised December 24, 2021.
Manuscript publicized February 16, 2022.
† The authors are with the Tokyo Metropolitan University, Hinoshi, 191-0065 Japan.
a) E-mail: hirakura-naoki@ed.tmu.ac.jpb) E-mail: aida@tmu.ac.jpc) E-mail: k-kawa@cc.tuat.ac.jp
DOI: 10.1587/transcom.2021EBP3150

tions to social media posts based on the following ideas.
People are more likely to accept others’ opinions if they
support their preferred opinions, and to ignore the opposing
opinions. This is called confirmation bias [6]. In particular,social media promotes this, making it easy to form clusters ofusers with similar opinions. In a closed community of people
with similar opinions, the same opinions are reinforced, anddifferent opinions are actively suppressed. This can create a
special kind of common sense that is valid only within a community; it is referred to as the echo chamber phenomenon.
For example, biased linkage patterns in Blogs about political
topics [7] and social media users’ tendency clustering along
party lines [8] were found. Besides, quantifying echo chambers on Twitter that emerged with the impeachment of theformer Brazilian President was conducted [9]. These studies
support the presence of echo chambers in OSNs.
Moreover, people can feel strong repulsion towards
opinions that are different from their own. This means thatwhen people see an opinion that differs from their own, they
will come to believe more strongly in the opinion they originally had. In psychology this is called the backfire effect [10],and there is a report that confirms the backfire effect in social
media users [11].
A opinion formation model incorporates the backfire
effect was proposed [12]. This model is called the BEBA
model because it incorporates two types of interactions:backfire effect and biased assimilation. In this model, regardless of the actual distribution of opinions, neutral opinions
are artificially introduced in advance and are used as a standard to determine which interaction predominates. Based on
the assumption of neutral opinions, the BEBA model posits
that biased assimilation is established between users who
have the same opinions, while the backfire effect emerges
if the users have different opinions. Polarization occurs between users who have conflicting opinions, which are opinions on opposite sides of the neutral opinion. Polarization
never occurs if all users have opinions on the same side of theneutral opinion. Of importance, our paper proposes a modelthat reflects the idea that opinion neutrality is determined byrelative differences in user opinions.
In this paper, we propose a model of user opinion formation as evidenced by interaction via social media. A uniquefeature of our model is that it eliminates the need to establish
an absolute opinion neutrality point. This feature reflects
our idea that polarization arises with relative differences inuser opinions, rather than from any predetermined neutral
opinion. Our model is designed to change continuously the

Copyright © 2022 The Institute of Electronics, Information and Communication Engineers

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

991

strength of the interaction based on differences in opinion.
Also, two types of reactions to social media posts are considered: empathy and repulsion. We assume that the former isstronger than the latter as opinions are closer, and its strength
decays as opinion difference strengthens. Conversely, the latter is stronger than the former when the opinions differ, andits strength decays the closer the opinions are.
To understand the behavior of the proposed model, we
conduct simulations. They reveal parameter characteristics
and show that the model can form either large groups or
multiple small groups.
The rest of this paper is organized as follows. In Sect. 2,
we explain related work and clarify the position of this study.
Section 3 proposes a model of opinion formation based on
empathetic and repulsive reactions to social media posts.
In Sect. 4, we evaluate the proposed model and show how
different parameters contribute to polarization. Section 5compares the proposed model with previous study and showsthat it can effectively deal with polarization. In Sect. 6, weconclude.
2.

Related WorkAs mentioned in Sect. 1, features of the proposed model arethat it obviates the need to artificially introduce the neutral
point of opinions, that the strength of the interaction changes
based on the difference in opinions, and that the opinion
value is updated based on information diffusion on the OSN.
We will explain the key advances over existing study by
discussing these three points.
First, we discuss the fact that the proposed model eliminates the need to artificially introduce a neutral point of
opinion in advance. In addition to [12], which we discussed
in Sect. 1, [13] is an existing study that introduces repulsive
interactions after assuming that the objective neutral opinion
is 0. In that study, it is rare for agents to completely change
their original opinion by crossing over the neutral point of
opinion. The interactions of users with opinions of different
signs are governed by different rules. For the situation inwhich the two agents’ opinion values are close but their signis different, [13] proposed three types of interactions: negative interactions, indifference, and repulsion. Regardless ofinteraction type, the sign of each agent’s opinion value does
not change. Besides, many other models also artificially introduce a neutral point of opinion in advance at the originof the axis. Typical models that reflect such thought about
opinion neutrality are DeGroot model [14] and FriedkinJohnsen (FJ) model [15]. Ordinary, these kinds of models
describe the user’s opinion as continuous real value in [−1, 1]
or in [−∞, +∞], and 0 represents a neutral opinion. Manymodels that extend DeGroot model and FJ model have been
proposed. For example, there are extended models focusing
on polarization: a model based on biased assimilation that
extends the eGroot model [16] and a model examining thefilter bubble effect to the polarization that extends the FJ
model [17]. Since each model extends the DeGroot modelor the FJ model, thought of opinion neutrality is the same.

There are also models that have the same thought of opinion neutrality other than DeGroot, FJ, and these extending
models. For example, [18] use a sign of the opinion value todescribe difference of user’s stance. On the other hand, since
our model sets no objective neutral opinion and instead addresses the difference in opinion strength, it allows for more
subtle interactions among users.
Next, we discuss how the strength of the interactionchanges based on the difference in opinions. The bounded
confidence model is a model that determines the presence orabsence of an interaction based on the difference in opinions.
This model uses a predetermined threshold value, called
the confidence radius, and interaction occurs only between
agents whose difference in opinion value is less than or equal
to the confidence radius. One type of bounded confidence
model is the Deffuant-Weisbuch (DW) model [19]. In thismodel, if the difference in the opinion values of two randomly
selected agents is less than or equal to the confidence radius,
their opinions approach each other.
Another type of bounded confidence model, theHegselmann-Krause (HK) model, updates opinion values
synchronously [20]. In this model, if the difference in theopinion values of two agents is less than the confidence radius, it is considered that they are adjacent on the network.
The user’s opinion value is updated to the average value ofthe opinion values of itself and neighboring users.
In addition, a model that extends the HK model basedon social judgment theory has been proposed [21]. In thismodel, one agent classifies another agent as being suitable for
acceptance, non-commitment, or rejection based on the difference in opinion; the corresponding responses are positive
acceptance, ignoring, and negative perception. Two thresholds are needed to establish these three judgement types.
As shown above, in the bounded confidence model andits extended model, threshold values are set in advance todetermine the existence and type of interaction. Whereas
these previous studies set thresholds in advance and so lackflexibility, this paper proposes a continuous, threshold-lessmodel.
In addition, the proposal differs in terms of how it handles repulsive interactions. In [21], based on the social
judgment theory, agents whose opinions are close to eachother always develop a positive attitude, and agents who
have different opinions always develop a negative attitude.
The proposed model, on the other hand, stochastically selects empathetic and repulsive reactions, and the strength of
the reaction is determined by a continuous function of thedifference in opinion values: if the opinions are close, theempathy is large and the repulsion is small. It should be
noted that the amount of change in the opinion value withrepulsion when opinions are close is so small that there is virtually no substantially change in opinion. The same can besaid for empathy and repulsion when opinions are far apart.
The strength of the empathetic and repulsive reactions will
be similar for those who are some distance apart in opinion,
and both reactions are moderate.
Similar to the proposed model, a model that defines

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

992

interaction rules as a continuous function is also proposed,in which the opinions of users approach each other if they
are close, and they move away from each other if they are farapart [30]. In [30], the cubic curve is used as the simplest
form to represent the reaction, which is attractive when theopinions are close and repulsive when they are far apart.
Although the approximation of cubic functions is a main
term of the Taylor expansion near the equilibrium point, it isunnatural to define it in the whole parameter range. On theother hand, the proposed model assumes that the strength of
the reaction changes exponentially. In the proposed model,
the strength of the interaction is reasonable. For example,
the rate of decrease in the strength of the interaction when the
difference increases by δ is exp(−k(|∆| + δ))/exp(−k |∆|) =exp(−kδ), which is constant and highly universal regardlessof the value of δ.
In [21] and [30], opinions are definitely attracted witheach other when opinions are close and opinions are definitely repulsed each other when opinions are far away. On
the other hand, the proposed model probabilistically chooses
empathetic reaction or repulsive reaction. Though the average characteristics of the proposed model are similar to themodel [30], social dynamics are not always driven by average characteristics. Individual reactions are diverse andsuch fluctuation of individuals can influence macro social
dynamics. Our proposed model can describe individual reactions with fluctuations. It is necessary to compare how
the difference between the two models actually affects the
dynamics, but this is future work.
Next, we discuss how to update opinion values to reflect information diffusion over the OSN. This is related tothe choice of interact partners. In this paper, opinion formation is considered to occur with information diffusion on
OSNs. Existing opinion formation models describe several
ways in which interaction partners are selected: all agents
update their opinions synchronously [14], [15], or two randomly selected agents update their opinions [19], [22]. In
this paper, the multivariate Hawkes process is used to determine the user who posts and the time of the posting, andthe posting user’s neighbors may update their opinions in response. The multivariate Hawkes process is a point processwith mutual excitation in which multiple processes promote
the occurrence of each other’s events [23]. It is effective inmodeling posts on social media, where a post by one user
triggers posts by its neighbors [24], [25].
This paper is an extension of our prior study [26]. Ina specific advance, we evaluate the behavior of the model
when each user has different parameters.
3.

Modeling Polarization by Empathy and Repulsion

We model situations in which a user’s posts on social media
influence the opinions of other users. In this model, two typesof interactions are considered: empathy and repulsion. Users
strongly empathize with posts that are close to their opinions
and dismiss posts that express opinions very different fromtheir own. In the case of repulsion, users are repulsed more

Fig. 1
action.

Amounts of opinion changing by empathetic and repulsive inter-by posts that are more distant from their opinions, and areless likely to be repulsed by posts that are closer to their own
opinions. These two kinds of interactions are stochasticallychosen, therefore, three types of reactions are introduced:
empathy, repulsion, and disregard. Important points of the
proposed interaction rule are that it eliminates the need to
introduce both a point of neutral opinion and a thresholdthat determines the type of interaction. Intuitive drawingof interaction rules is shown in Fig. 1. When the difference
between two users’ opinions is small, the amount of opinionchanging by empathetic interaction is large and the amount of
opinion changing by repulsive interaction is tiny. Conversely,
when the difference between two users’ opinions is large, theamount of opinion changing by empathetic interaction is tiny
and the amount of opinion changing by repulsive interaction
is large. The following rules determine how opinions arechanged based on these reactions.
Consider a social network with N users. User i (i =1, . . . , N) is deemed to have opinion value oi (t) ∈ [−1, 1];p
the opinion value oi (t) of user i towards a post at time t ispoi (t) = oi (t) and reflects the user’s opinion at that time. We
assume that the opinion of user i is impacted by concurrent
posts. If user j is the author of the latest post seen by useri, then the opinion value of user i at time t changes in rep
sponse to o j (t − ), where t − represents time right before t. The
opinion value of user i is given by oi (t) = oi (t − ) + f (oi (t − ),po j (t − )). The second term on the right side is explained later.
When user i sees the latest post of user j, user i develops
either an empathetic or repulsive reaction with probability ofpi or 1 − pi , respectively. We propose a model of change inopinion value, assuming that the magnitude of the empathy
and repulsion depends on the difference in opinion values.

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

993

Fig. 2

p

These two illustrations show the amount of opinion change in the cases of ∆ = o j (t − )−oi (t − ) ≤p
0 (Fig. 2(a)) and ∆ = o j (t − ) − oi (t − ) > 0 (Fig. 2(b)). The cyan circles and orange squares are the user’sand post’s opinion values, respectively. The blue arrow and red arrow indicate the amount of opinionchanging by empathetic reaction and repulsive reaction, respectively.

First, consider the case of empathy. User i should show
a stronger empathetic reaction to posts whose opinion values
are close to their own, and weaker empathy when the opinion values are very different. Based on this, we introducepfunction f (oi (t − ), o j (t − )), which represents the change inthe opinion value, as follows∆ c k exp(−k |∆|).

(1)pNote that ∆ = o j (t − ) − oi (t − ) is the difference between theopinion value of the latest post seen by user i and the user’s
opinion value. Parameter k(> 0) is a constant that determines the attenuation in the intensity of empathy with respect
to the absolute value of the difference of opinion values |∆|,and c is a parameter that is adjusted to yield oi (t) ∈ [−1, 1],
which satisfies ck ≤ 1.
Next, we consider the case of repulsion. In this case,a stronger repulsive reaction should be developed to posts
that are very different from own opinion values, and therepulsion should weaken as the opinion values approach eachother. In order to express the repulsive reaction in the same
form as empathy, we impose a periodic boundary conditionon the two ends of +1 and −1 to harmonize the difference
between the opinion value of a post and the user’s opinion,|∆0 | = 2 − |∆|; the smaller the difference |∆0 | is, the larger
the repulsive reaction becomes. Based on this, we designpfunction f (oi (t − ), o j (t − )), which represents the change inopinion value, as(1 − oi (t − )) c k exp(−k |∆0 |) , (∆ ≤ 0),− (1 + oi (t − )) c k exp(−k |∆0 |) , (∆ > 0).

(2)
(3)

Figure 2 illustrates the concrete responses of expressions (1), (2), and (3). The blue circle indicates a user’s
opinion and the orange square indicates the post’s opinion.
Numbered line of [−1, 1], the solid line, delineates opinion
space. We also impose a periodic boundary condition on thetwo ends of +1 and −1 to define the new difference of theopinion values as |∆0 | and the relationship is |∆0 | = 2− |∆|. It
is used to define the opinion change rule of the repulsive reaction in the same form as the empathetic reaction. Note thatthe orange square on the dashed line indicates the same opinion on the solid line. The green arrow indicates the amountof opinion value change caused by the empathetic reaction.
The smaller |∆| is, the stronger the empathy is. The purple
arrow indicates the amount of opinion value change caused
by a repulsive reaction. The smaller |∆0 | is, the stronger the
repulsion is.
Expression (1) shows the change in the opinion value inthe case of an empathetic reaction; it indicates that the user’s
opinion value approaches the opinion value of the post by
an amount proportional to the difference, c k exp(−k |∆|). In
this case, the positive or negative value of ∆ corresponds tothe direction of the movement in the user’s opinion to becloser to the opinion of the post as shown in the negative
direction of Fig. 2(a) and the positive direction of Fig. 2(b).
Figure 2(a) shows the case of ∆ ≤ 0. The arrow pointing in the negative direction from the blue circle representing the user’s opinion value indicates that the user’s opinion
value approaches the opinion value of the post by the proprtion c k exp(−k |∆|) of difference |∆|. On the other hand,Fig. 2(b) shows the case of ∆ > 0, where the sign of ∆

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

994

denotes a positive change in opinion value in the opposite
direction to that of Fig. 2(a).
Expressions (2) and (3) show the change in opinion
value under a repulsion reaction. First, we will discuss thecase of ∆ ≤ 0. As shown in Fig. 2(a), the opinion valueof user i increases by the proportion c k exp(−k |∆0 |) of difference 1 − oi (t − ) between the opinion value of user i andthe most extreme opinion value. In this case, the amountof change in the opinion value becomes the expression (2)and the user’s opinion value moves away from the posted
opinion value. We now discuss the case of ∆ > 0. As shownin Fig. 2(b), the opinion of user i decreases by the proportionof c k exp(−k |∆0 |) of difference −(1 + oi (t − )) between theopinion value oi (t − ) of user i and the most extreme opinion−1. In this case, the amount of change in the opinion value
is given by the expression (3), where the user’s opinion value
moves away from the posted opinion value.
Parameter k (> 0) represents the attenuation rate of thestrength of the effect, and the value of k characterizes the rateof change in the strength of the effect based on the differencein opinion values. Parameter k allows us to design opinion
value change rules where users show a stronger empathetic
reaction to posts with close opinion values and a stronger
repulsive reaction to posts with distant opinion values.
The limitation of the proposed model is that the rangeof the opinion values can only be a finite interval. This isbecause it is not possible to define periodic boundary conditions. However, this limitation can be relaxed by introducing
an appropriate scaling. The typical scaling is stereographic
projection, which gives a bijection between points in (−∞, ∞)and points on an open section on a circle with a finite radius.
4.

Evaluations of the Proposed Model

4.1

Preparation for Evaluations

We explain how we generated sequences of posts and theindex of polarization used in the evaluations of the proposedmodel.
4.1.1

Generating a Sequence of Posts

Here, we employ the multivariate Hawkes process to generate artificial data [23]. The multivariate Hawkes process isa point process in which multiple processes mutually excite
the occurrence of events. To use this for the generation ofuser post sequences on social media, we assume that eachprocess is considered to be a user and each event is a post tosocial media, and that posts are promoted between connected
users.
The posting rate, λi , of user i (i = 1, . . . , N) is expressed as follows.
Õ Õ(4)
λi (t) = µi +α ji e−β j i (t−th ),j ∈∂i th ∈H j

where µi is the base rate, ∂i is the set of neighbors of useri, H j is the set of past posting times of user j, and th is its
element. α ji and β ji ( j = 1, . . . , N, i = 1, . . . , N) represent
the rate jump and attenuation rate for user i due to user
j’s posts, respectively. Thus, the posting rate of user i isdetermined by the user-specific base rate and the previous
posts of neighboring users.
We generate a sequence of posts using the multivariate
Hawkes process. First, social networks with 100 nodes are
created by the Barabási-Albert model (hereinafter referred
to as the BA model) [27] and complete graph. In Eq. (4), α ji
and β ji are given as uniform random numbers in the rangeof [0, 1] for each combination of j and i ( j, i = 1, . . . , N).
The base rate, µi , is given as a uniform random number inthe range of [0, 1]. The following experiments use the 10000
postings generated by the multivariate Hawkes process.
Simulations are conducted according to thinning
method [29]. This method is briefly explained in Appendix.
4.1.2

Index of Polarization

We introduce the index [28] for frequency distribution(π, y) = (π1, . . . , πn ; y1, . . . , yn ) as the index of polarization:
P=K

n Õ
n
Õ

πi1+θ π j yi − y j ,(5)

i=1 j=1

where n is the number of classes that equally divide theopinion value space [−1, 1], yi is the i-th class value fromthe bottom, πi is the number of users belonging to the i-thclass, K is a parameter for normalization, and θ is a parameter
called the polarization sensitivity, which takes a value in therange of (0, θ ∗ ' 1.6]. According to [28], the stronger thesense a user has of belonging to a class and the greater
the degree of hostility toward another class, the greater isthe degree of polarization. The index of polarization (5) is
designed to satisfy the appropriate axioms for a valid index ofpolarization. For example, the maximum value is achieved if
half of the users belong to the lowest and half to the highest
class, and the minimum value is taken if all users belong to
the same class.
It is often observed that users are divided into groups
with conflicting opinions and slander each other on OSNs.
The polarization index (5), which has a higher value when
the group size is larger and the opinions between groups are
farther apart, is considered to be effective in evaluating thepolarization of people’s opinions.
In the following evaluation experiments, we set the parameter θ = 0.5 and the number of classes to 10 for thefollow reasons. This index can quantitatively measure thedegree of polarization in the distribution of opinion values
if the parameters used to identify the degree of polarizationof each opinion value distribution are properly set.
If the value of parameter θ, called the polarization sensitivity, is small, the value of the polarization index P will not
be large unless the bias of the distribution is large. However,if the sensitivity to polarization takes a value of (0, θ ∗ ' 1.6],

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

995

Fig. 3 Examples of opinion value change conducted in BA graph. These are typical patterns of opinion
formation and these indices of polarization are as follows: (a) low, (b) high, and (c) middle.

Fig. 4

Changing of polarization index corresponding to Fig. 3.

it can work as an indicator of polarization. Also, if the number of classes is too small, such as 2 or 3, there will be nodifference among the users belonging to each class in the
distribution of different opinion values, and the polarizationindex is likely to have a high value even if the opinion values
are randomly distributed. The polarization index works well
as long as the number of classes is not set to such extreme
settings.
4.2

Parameter Characteristics of the Proposed Model

We conduct evaluation experiments to elucidate the characteristics of the parameters of the proposed model. The
evaluation conditions are as follows. The initial opinion is
given as a uniform random number in [−1, 1]. We apply theopinion value change rule to the sequence of posts generated
in Sect. 4.1.1, with the probability of an empathetic reaction
and the probability of a repulsive reaction being pi = p and
1 − pi = 1 − p, respectively, for all i.
Three typical simulation results with different parameters for the opinion change using the proposed model inthe case of BA graph are shown in Fig. 3 and the case ofcomplete graph are shown in Fig. 5. Besides, the changesof polarization index corresponding to opinion changing areshown in Fig. 4 and Fig. 6, respectively.
We discuss the simulation results for BA graph first.
The probability of empathy p is set to p = 0.9 and p = 0.1
for the cases of Figs. 3(a) and 3(c), respectively; parameterk is fixed at 2. The difference in the probability of empathy

reaction p yields the consensus of opinions in Fig. 3(a) andthe polarization of opinions in Fig. 3(b). Figure 3(c) shows
the case of k = 10. Since the influence of other users’ posts
is smaller than when k = 2, the opinion values of each user
do not change much from their initial values. Some users
did not join large groups in 3(a), 3(b), and 3(c). This isbecause the number of interactions is different for each user
by using a sparse network model, the BA graph. AlthoughFig. 3(c) does not show complete polarization, the simulationis considered to be converged, as can be seen from the factthat the polarization index in Fig. 4(c) does not change much.
Even if the polarization progresses further from this state,
there are many users whose opinions do not change due tothe influence of the graph structure, so the polarization index
does not fluctuate significantly to reach the maximum value.
Next, we discuss simulation results for a completegraph. Experimental conditions are same as simulationsof BA graph; the parameter k is fixed at 2 and the probabilityof empathy p is set to p = 0.9 and p = 0.1 for the cases ofFigs. 5(a) and 5(b), respectively. The results are similar tothe case of the BA graph, where consensus and polarization
occur due to differences in p. Figure 5(c) shows the casewhere the parameters are p = 0.9 and k = 10. Two groups
are formed, but each group does not have the opinion value±1. When k takes a relatively large value, it strongly empathizes only with users whose opinions are very close andstrongly repels only with users whose opinions are far apartand hardly influence each other in the middle. Therefore, thetwo groups have little influence and converge to an opinion

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

996

Fig. 5 Examples of opinion value change conducted in complete graph. These are typical patterns of
opinion formation and these indices of polarization are as follows: (a) low, (b) high, and (c) middle.

Fig. 6

Changing of polarization index corresponding to Fig. 5.

value other than ±1.
Comparing the case of BA graphs (Fig. 3) with the caseof complete graphs (Fig. 5), the difference is that the convergence is faster in the case of complete graphs, and all users
belong to the group. This is because in the case of BA graph,
the number of interactions is biased for each user due to the
non-uniformity of node degree, and in the case of completegraphs, anyone’s posts affect all other users.
We then evaluate the index of polarization of the final
state at the end of simulation using the index of polarization (5), for each combination of parameters p and k. The −(2+θ)Ín
normalization parameters are set to K = i=1πi, thepolarization sensitivity is set to θ = 0.5, and the number ofclasses is set to n = 10. Thus the possible values of thepolarization index lie approximately in the range [0, 0.64).
We conducted 30 experiments with different initial
opinions and evaluated the probability of the consensus
reached and the probability of the polarization attained. Consensus reached is defined as satisfying P ≤ 0.064 at the endof the simulation, and polarization attainment is defined assatisfying P ≥ 0.576 (top and bottom ten percent of thepossible values of the polarization index [0, 0.64)).
Figure 7 shows the probability of consensus reached foreach combination of p and k. When k was relatively large,
consensus was not reached. This is because empathy decays
more strongly when k is high. When k is relatively small,Fig. 7 Consensus probability of each combination of parameters. We
conducted 30 simulations for each combination of p and k in BA graph,and evaluated the percentage of users reaching consensus (P ≤ 0.064).

the probability of consensus depends on probability p. The
closer p is to 1, the higher in the probability of consensus.
Figure 8 shows the probability of polarization for eachcombination of p and k. When k is relatively large, nopolarization occurs. This is the same as in the probabilityof consensus since users are less influenced by differing
opinions on posts given relatively large k values. When k isrelatively small, the probability of polarization depends on

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

997

Fig. 8 Polarization probability of each combination of parameters. We
conducted 30 simulations for each combination of p and k in BA graph,and evaluated the percentage of polarization (P ≥ 0.576).

Fig. 10 Polarization probability of each combination of parameters. We
conducted 30 simulations for each combination of p and k in completegraph, and evaluated the percentage of users reaching consensus (P ≥0.576).

4.3

Fig. 9 Consensus probability of each combination of parameters. We
conducted 30 simulations for each combination of p and k in completegraph, and evaluated the percentage of users reaching consensus (P ≤0.064).

the probability, p, of an empathetic reaction. In contrast to
the probability of consensus, polarization probability rises
as p approaches 0.
We also investigated parameter characteristics in a complete graph. Figure 9 shows the consensus rate for the caseof complete graphs. The characteristic that the consensus
rate is high when the parameter k is relatively small and theempathy probability p is relatively large is the same as in
the case of the BA graph. However, there is a difference
that the consensus rate is higher even in regions where k islarger. Figure 10 shows the polarization rate for the complete graph. As in the case of the BA graph, the polarization
rate is high when the parameter k is relatively small and theempathy probability p is relatively small. However, there isa difference that the polarization rate is high even in regions
where k is larger. The reason for this difference is due to thestructure of BA graphs and complete graphs, as described in
the comparison between Fig. 3 and Fig. 5.

Model Behavior under Heterogeneous Parameter Combinations

In this section, we conduct simulations under the condition
that the parameters are different for each user. The specificconditions of the experiment are as follows. We prepare a
network with 50 users and each user’s parameter k is set
to k = 5 or k = 15. We simulate using different settingsof mixing ratio of the population of users with k = 5 andk = 15. Specifically, we conduct experiments with eachcombination of the following: (a) 50 users with k = 5, (b)40 users with k = 5 and 10 users with k = 15, (c) 30 users
with k = 5 and 20 users with k = 15, (d) 25 users with k = 5
and 25 users with k = 15, (e) 20 users with k = 5 and 30
users with k = 15, (f) 10 users with k = 5 and 40 users
with k = 15, and (g) 50 users with k = 15. We conduct
100 simulations and compare the distribution of the index ofpolarization at the end of each simulation for the different
mixing ratios. The probability of an empathetic reaction isp = 0.5 for all users. Parameters and the number of classesof polarization index (5) are the same settings used for theexperiments in Sect. 4.2.
Figure 11 shows the distribution of polarization at theend of the simulation for each mixture ratio with different
values of parameter k. In Fig. 11(a), the parameter k = 5
for all users. Then, the lowest class of polarization index isthe most frequent and the highest class is the second mostfrequent. The reason why the consensus is often achieved isthat the attenuation in the strength of empathy k is relatively
small. However, if the opinions are too far apart, the strengthof empathy decreases, and the effect of repulsion increases.
Once users start to get close to each other, they often reachconsensus, whereas interactions between users with quite
different opinions lead to polarization. As the mixing ratioof the population of k = 5 and k = 15 changes, the number ofcases taking lower and higher polarization decreases, and thenumber of cases taking middle polarization index increases.

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

998

Fig. 11 Frequency distribution of the polarization index at the end of the simulation under mixed
conditions of users with different parameters. In Fig. 11(a), all users’ parameters are k = 5. In the orderof Fig. 11(b), 11(c), . . . , 11(g), the ratio of users with k = 5 decreases and the ratio of users with k = 15
increases. When the ratio of k = 5 is higher, it is easy to form a large group so that the polarization index
takes lower and higher values. As the ratio of k = 15 increases, users tend to form multiple small groups
so that the polarization index takes middle values. Besides, dashed lines are threshold to determine
consensus and polarization.

Users with a relatively large attenuation in empathy strength,k, are more empathetic to posts that are very close to theirown but are less influenced by opinions slightly further apart.
As the number of users with relatively large k increases,
the variability of the opinion values in the whole network
decreases, and middle polarization index cases were more
frequent at the end of the simulation. From this result, it canbe seen when the proportion of users with k = 5 is large,
it is easy to form a large population while an increasing
proportion of users with k = 15 tends to form multiple small
groups.
5.

Comparison with Previous Research

We compare our model with the opinion formation model,
which takes into account the reactions corresponding to em-

pathy and repulsion. As an example, we take the BEBA
model [12]. The BEBA model is based on the classicalopinion formation model of DeGroot model [14]; BEBA
contains biased assimilation and the backfire effect. Biased
assimilation means that we highly evaluate information thatis consistent with our original opinion, and the backfire effectmeans that when we are exposed to an idea that is different
from our own, we come to believe more strongly in our
original opinion.
The BEBA model is briefly described as follows. First,
consider a social network with N users. User i (i = 1, . . . , N)
has opinion value yi (t) ∈ [−1, 1] at time t. For adjacent users
i and j, if opinion values yi (t) and y j (t) are the same sign,biased assimilation dominates, but if they are different signs,
backfire effect dominates. This means that opinion value
0 is neutral and users have different opinions depending on

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

999

opinion values was determined based on the difference inthe opinion values, so that repulsion occurred and polarization was created. Depending on the parameter values used,
the proposed model was found to yield cases of polarizationin which repulsive reactions were active even with a biased
opinion value distribution.
In the case of a biased initial distribution, such that allusers’ opinion values are the same sign, the BEBA model
always reaches consensus. Besides, [13] also predetermines
the opinion neutrality and the sign of opinion value neverchanges. On the other hand, the proposed model can predictcases of polarization depending on the parameter settings.
Also, the proposed model can handle different cases of opinion change, including cases where a consensus is reached
if we set different parameters. The difference in the results
between the two models is due to the difference in the ideaof opinion neutrality. Note that this comparative experimentis only conducted with BEBA model but other models thatneed to artificially introduce opinion neutrality in advancebasically behave in a similar way to BEBA model. As wediscussed in Sect. 2, other models that artificially introduce
opinion neutrality in advance include [14]–[18]. The proposed model reflects the idea that opinion neutrality is not
artificially introduced but by the relative difference in users’
opinions; it can handle a wider range of opinion changecases.
6.

Fig. 12 Change in opinion values under the condition such that all users’
opinion values have the same sign.

whether they have positive or negative opinion values. On
the other hand, in the proposed model, the amount of change
in the opinion value is determined based on the difference
between the post’s and user’s opinion values. In order toclarify the difference between the two models, we set all
initial opinion values to have the same sign in a comparativeevaluation.
Figure 12 shows comparison of experimental results of
BEBA model and the proposed model. First, we describe theresults of the experiments on the BEBA model. Figure 12(a)shows the change in opinion values yielded by the BEBA
model. Since the initial opinion values with the same sign
were given to all nodes, the backfire effect was quiescent,and the opinions became concentrated at the end of the simulation. Next, we describe the results of the experiments on
the proposed model. Figure 12(b) shows the change in opinion values for p = 0.1 and k = 2 yielded by the proposedmodel. Although the initial opinion values with the same
sign were given to all nodes, the amount of change in theConclusion

This paper proposed a model to simulate polarization of social media users. The model is characterized by two typesof reactions: empathy and repulsion, with different strengths
of influence being created by differences in user opinion
values. It also reflects the idea that relative differences inopinion determine the neutrality of opinion. In particular,
since our idea of opinion neutrality dispenses that the artificially introduced concept of opinion neutrality, it has theadvantage of being able to respond flexibly to biased distributions of opinion values. These features allow a wide range
of user characteristics to be appropriately modeled so as towell cover the phenomenon of polarization. The dependenceof the change in the opinion value on the initial values of theopinions will be examined in the future.
Acknowledgments
This research was supported by Grant-in-Aid for Scientific
Research (B) No. 19H04096 (2019–2021), No. 20H04179(2020–2022), and No. 21H03432 (2021–2023), andGrant-in-Aid for Challenging Research (Exploratory)
No. 21K19775 (2021–2023) from the Japan Society for the
Promotion of Science (JSPS), and TMU local 5G researchsupport.
References[1] C.R. Sunstein, “#Republic: Divided Democracy in the Age of Social

IEICE TRANS. COMMUN., VOL.E105–B, NO.8 AUGUST 2022

1000

Media,” Princeton University Press, 2018.
[2] W.J. Brady, J.A. Wills, J.T. Jost, J.A. Tucker, and J.J. Van Bavel,“Emotion shapes the diffusion of moralized content in social networks,” National Academy of Sciences, vol.114, no.28, pp.7313–
7318, 2017.
[3] I. Weber, V.R.K Garimella, A. Batayneh, “Secular vs. Islamist polarization in Egypt on Twitter,” IEEE/ACM International Conference
on Social Networks Analysis and Mining (ASONAM), pp.290–297,
2013.
[4] A.V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysis of dynamic social networks. Part II,” Annual Reviews in Control,
vol.45, pp.166–190, 2018.
[5] M. Jalili and M. Perc, “Information cascades in complex networks,”
Journal of Complex Networks, vol.5, no.5, pp.665–693, 2017.
[6] R.S. Nickerson, “Confirmation bias: A ubiquitous phenomenon in
many guises,” Rev. Gen. Psychol., vol.2, no.2, pp.175–220, 1998.
[7] L.A. Adamic and N. Glance, “The political blogosphere and the 2004
US election: Divided they blog,” Proc. 3rd international workshop
on Link discovery, pp.36–43, 2005.
[8] M.D. Conover, B. Gonçalves, A. Flammini, and F. Menczer, “Partisan asymmetries in online political activity,” EPJ Data science, vol.1,
p.6, 2012.
[9] W. Cota, S.C. Ferreira, R. Pastor-Satorras, and M. Starnini, “Quantifying echo chamber effects in information spreading over politicalcommunication networks,” EPJ Data Sci., vol.8, p.35, 2019.
[10] B. Nyhan and J. Reifler, “When corrections fail: The persistence ofpolitical misperceptions,” Polit. Behav., vol.32, no.2, pp.303–330,
2010.
[11] C.A. Bail, L.P. Argyle, T.W. Brown, J.P. Bumpus, H. Chen,
M.F. Hunzaker, J. Lee, M. Mann, F. Merhout, and A. Volfovsky,“Exposure to opposing views on social media can increase politicalpolarization,” Proc. National Academy of Sciences, vol.115, no.37,
pp.9216–9221, 2018.
[12] X. Chen, P. Tsaparas, J. Lijffijt, and T. De Bie, “Opinion dynamics with backfire effect and biased assimilation,” arXiv preprint
arXiv:1903.11535, 2019.
[13] C. Altafini and F. Ceragioli, “Signed bounded confidence models for
opinion dynamics,” Automatica, vol.93, pp.114–125, 2018.
[14] M.H. DeGroot, “Reaching a consensus,” J. Am. Stat. Assoc., vol.69,
no.345, pp.118–121, 1974.
[15] N.E. Friedkin and E.C. Johnsen, “Social influence and opinions,”
Journal of Mathematical Sociology, vol.15, no.3-4, pp.193–206,
1990.
[16] P. Dandekar, A. Goel, and D.T Lee, “Biased assimilation, homophily,
and the dynamics of polarization,” Proc. National Academy of Sciences, vol.110, no.15, pp.5791–5796, 2013.
[17] U. Chitra and C. Musco, “Analyzing the impact of filter bubbles onsocial network polarization,” Proc. 13th International Conference on
Web Search and Data Mining, pp.115–123, 2020.
[18] F. Baumann, P. Lorenz-Spreen, I. Sokolov, and M. Starnini, “Modeling echo chambers and polarization dynamics in social networks,”
Phys. Rev. Lett., vol.124, no.4, p.048301, 2020.
[19] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, “Mixing beliefs
among interacting agents,” Adv. Complex Syst., vol.3, no.01n04,
pp.87–98, 2000.
[20] R. Hegselmann and U. Krause, “Opinion dynamics and bounded
confidence models, analysis, and simulation,” Journal of Artificial
Societies and Social Simulation, vol.5, no.3, pp.1–33, 2002.
[21] H.D. Aghbolagh, M. Zamani, S. Paolini, and Z. Chen, “Balanceseeking opinion dynamics model based on social judgment theory,”
Physica D: Nonlinear Phenomena, vol.403, Art. no.132306, 2020.
[22] G. Deffuant, F. Amblard, G. Weisbuch, and T. Faure, “How can extremism prevail? A study based on the relative agreement interaction
model,” Journal of Artificial Societies and Social Simulation, vol.5,
no.4, 2002.
[23] A.G. Hawkes, “Spectra of some self-exciting and mutually exciting
point processes,” Biometrika, vol.58, no.1, pp.83–90, 1971.

[24] M.A. Rizoiu, Y. Lee, S. Mishra, and L. Xie, “Hawkes processes forevents in social media,” Frontiers of Multimedia Research, C. ShihFu, ed., Chapter 8, pp.191–218, Association for Computing Machinery and Morgan & Claypool, Cham, 2017.
[25] A.H. Zadeh and R. Sharda, “Hawkes point processes for social media
analytics,” Reshaping Society through Analytics, Collaboration, andDecision Support, L.S. Iyer, D.J. Power, eds., Chapter 5, pp.51–66,
2015.
[26] N. Hirakura, M. Aida, and K. Kawashima, “A model of polarization
on social media caused by empathy and repulsion,” arXiv preprint
arXiv:2011.08141, 2020.
[27] A.-L. Barabási and R. Albert, “Emergence of scaling in random
networks,” Science, vol.286, no.5439, pp.509–512, 1999.
[28] J.-M. Esteban and D. Ray, “On the measurement of polarization,”
Econometrica, vol.62, no.4, pp.819–851, 1994.
[29] Y. Ogata, “On Lewis’ simulation method for point processes,” IEEE
Trans. Inf. Theory, vol.27, no.1, pp.23–31, 1981.
[30] D. Sabin-Miller and D.M. Abrams, “When pull turns to shove: A
continuous-time model for opinion dynamics,” Phys. Rev. Research,
vol.2, no.4, p.043001, 2020.

Appendix:

Simulation Method of Multivariate Hawkes
Process for Generating a Sequence of Posting

We used the thinning method [29] for simulating themultivariate Hawkes process and each user have posting rate:Õ Õ
α ji e−β j i (t−th ) .
(A· 1)λi (t) = µi +j ∈∂i th ∈H jÍNUsing summation of all users’ posting rates, λ = i=1
λi ,we determine the time of next posting. Next, we determine
who post by ratio of posting rate λi /λ. Simulation detail is
described in Algorithm 1.
Algorithm 1 Simulation Method of Multivariate Hawkes
Process for Generating a Sequence of PostingÍNµi
1: λ∗ ← i=1∗
2: t ← 0
3: l ← 0
4: let T be the end time of simulation
5: while t ∗ < T do
6:
let E be a random number generated from exponential distribution
λ∗ exp(−λ∗ x)
7:let t ∗ ← t ∗ + E be a candidate of next posting time
8:
if t ∗ > T , terminate the simulation
9:
let r ← λ(t ∗ )/λ∗ be an adoption rate10:
let u be a random number generated from uniform distribution from0 to 1
11:
if u ≤ r then12:
adopt the posting time t ∗ as next posting time13:l ←l+114:tl ← t ∗15:
let v be a random number generated from probability distributionÍNP ∗ ← λi (t ∗ )/ i=1
λi (t ∗ ), thus v indicates posting user
16:
i←v
Í
17:
λ∗ ← λ(t ∗ ) + j ∈∂i α j i
18:
else
19:
reject the posting time t ∗ as next posting time20:
λ∗ ← λ(t ∗ )21:
end if
22: end while

HIRAKURA et al.: MODELING POLARIZATION CAUSED BY EMPATHETIC AND REPULSIVE REACTION IN ONLINE SOCIAL NETWORK

1001

Naoki Hirakurareceived his B.E. and M.E.
degrees in Engineering from Tokyo Metropolitan University, Japan, in 2018 and 2020, respectively. Currently, he is a Ph.D. student of
the Graduate School of Systems Design, Tokyo
Metropolitan University.

Masaki Aida
received his B.S. degree inPhysics and M.S. degree in Atomic Physics from
St. Paul’s University, Tokyo, Japan, in 1987 and1989, respectively, and his Ph.D. in Telecommunications Engineering from the Universityof Tokyo, Japan, in 1999. In April 1989, hejoined NTT Laboratories. From April 2005 to
March 2007, he was an Associate Professor at theFaculty of Systems Design, Tokyo Metropolitan University. He has been a Professor of
the Graduate School of Systems Design, Tokyo
Metropolitan University since April 2007. His current interests include
analysis of social network dynamics and distributed control of computercommunication networks. He received the Best Tutorial Paper Award andthe Best Paper Award of IEICE Communications Society in 2013 and 2016,
respectively, and IEICE 100-Year Memorial Paper Award in 2017. He isa fellow of IEICE, a senior member of IEEE, and a member of ACM and
ORSJ.

Konosuke Kawashimareceived his B.E.
degree from the University of Tokyo, Japan, in
1969, and Dr. Eng. degree from the same university in 1993. He is currently a Visiting Professor at Tokyo Metropolitan University, and a
Professor Emeritus at Tokyo University of Agriculture and Technology (TUAT). He joined NTT
Laboratories in 1969, where he engaged in the
research and development of teletraffic engineering for various networks and systems. From
1997, he was Director of Teletraffic ResearchCenter at NTT Advanced Technologies Corp. From 2002, he was a Professor at Department of Computer and Information Sciences of TUAT until
March 2012. He received the Young Engineer Award and Paper Award
from the IEICE in 1978, 1982 respectively. He also received the Best Paper
Award, Distinguished Achievement and Contributions Award from the Operations Research Society of Japan (ORSJ) in 1986, 2007 respectively, andthe Technical Award from the Telecommunications Advancement Foundation in 1996. He is a member of IEEE, and a fellow of ORSJ and IEICE.