2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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A Model of Polarization on Social Media Caused by Empathy andRepulsion
Naoki HIRAKURA†a) , Student Member, Masaki AIDA†b) , and Konosuke KAWASHIMA†c) , Fellows

SUMMARY In recent years, the ease with which social media can beaccessed has led to the unexpected problem of a shrinkage in information
sources. This phenomenon is caused by a system that facilitates the connection of people with similar ideas and recommendation systems. Bias in theselection of information sources promotes polarization that divides people
into multiple groups with opposing views and creates conﬂicts between
opposing groups. This paper elucidates the mechanism of polarizationby proposing a model of opinion formation in social media that considers
users’ reactions of empathy and repulsion. Based on the idea that opinion
neutrality is only relative, this model oﬀers a novel technology for dealing
with polarization.
key words: social media, polarization, ﬁlter bubble

1. Introduction
Today, social media services such as Twitter and Instagram
have become widely adopted and are playing a signiﬁcantrole in information distribution. Unlike traditional mass media, social media allows users to eﬃciently focus on views
that trigger their interest. Paradoxically, however, its very
power raises the problem of excluding most information
sources. This is because of the feature that allows users tochoose the accounts they chose to follow and the existence ofpush-style recommendation systems. It is feared that social
media will lead to polarization because it makes it easier
to connect people with similar opinions which biases theinformation distribution [1]. Polarization is the division of
opinion on a topic into basically two diametrically opposing
positions.
Polarization tends to emerge when users express theiropinions within a shuttered community; the opinions expressed are almost always seen as positive, with negative
opinions being drowned or pushed out. As a result, a special understanding that is common only within the community becomes strengthened, and a gap forms between thethoughts and values of people outside the community. This
reinforcement of belief within a community is called the echochamber phenomenon. Due to such bias in information distribution, slander, and defamation between communities thathave diﬀerent opinions become more frequent. To preventthese social problems, we should elucidate the mechanismof polarization on social media.
The basic reactions of users to posts on social media
are empathy and repulsion. The repulsive reaction is called† The authors are with Tokyo Metropolitan University, Hino,
Tokyo, 191-0065 Japan.
a) E-mail: hirakura-naoki@ed.tmu.ac.jpb) E-mail: aida@tmu.ac.jpc) E-mail: k-kawa@tmu.ac.jp

the backﬁre eﬀect in psychology [2], and one report has
conﬁrmed the backﬁre eﬀect of social media use [3].
Reference [4] proposed a model of opinion formationthat incorporates the backﬁre eﬀect. In this model, polarization does not occur automatically in the distribution of
biased opinions because the absolute origin is determinedby the quantitative representation of opinions. This model
corresponds to a situation where objective "neutral" positions are predetermined with regard to the distribution of
opinions. Polarization is assumed to occur between users
whose opinions are on the opposite sides of the neutral position.
In this paper, we propose a model that users are inﬂuenced by other users via social media such that their opinions
change. A feature of this model is the origin of the quantitative representations of opinions is not necessarily objective
neutrality. This feature reﬂects our view that opinion neutrality is relative. That is, polarization is generated from
relative diﬀerences in opinion, but not from the predetermined standard of neutrality. Also, the proposed model isdesigned to yield diﬀerent characteristics based on diﬀerences in opinion. There are two types of speciﬁc inﬂuence:
one is a strong empathy for posts that are close to one’s own
opinion and dismissal of diﬀerent opinions. The other is astrong repulsion for posts that diﬀer from their own opinion
and a ignoring of opinions that are close to their own.
The structure of this paper is as follows. In Section
2, we propose a model of opinion formation in which users
show empathetic and repulsive reactions to posts on social
media. In Section 3, we evaluate the model proposed in
Section 2 and clarify the behaviors of the model for given
parameters. Section 4 compares the proposed method withprevious works and shows the advantages of the proposed
method. We conclude in Section 5.
2.

Modeling Polarization by Empathy and Repulsion

We model situations in which a user’s posts on social media
inﬂuence the opinions of other users; two kinds of reactions
are considered, empathy and repulsion. When empathy occurs, users strongly empathize with posts that are close to
their opinions and dismiss posts that express opinions very
diﬀerent from their own. In the case of repulsion, users are
repulsed more by posts that are more distant from their opinions, and less likely to be repulsed by posts that are closer totheir own opinions. The following are the rules for changing
opinions based on these reactions.

Copyright © 2020 The Institute of Electronics, Information and Communication Engineers

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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Consider a social network with 𝑁 users. User 𝑖 (𝑖 =1, . . . , 𝑁) is deemed to have opinion value 𝑜𝑖 (𝑡) ∈ [−1, 1];
the opinion value 𝑜𝑖𝑝 (𝑡) of user 𝑖 for a post at time 𝑡 is𝑜𝑖𝑝 (𝑡) = 𝑜 𝑖 (𝑡) reﬂects the user’s opinion at that time. We
assume that the opinion of user 𝑖 is aﬀected by concurrent
posts. If user 𝑗 is the author of the latest post seen byuser 𝑖, then the opinion value of user 𝑖 at time 𝑡 changes
due to 𝑜 𝑝𝑗 (𝑡 − ). The opinion value of user 𝑖 is given by𝑜𝑖 (𝑡) = 𝑜𝑖 (𝑡 − ) + 𝑓 (𝑜𝑖 (𝑡 − ), 𝑜 𝑝𝑗 (𝑡 − )). The second term on theright side will be explained later.
When user 𝑖 sees the latest post of user 𝑗, user 𝑖 develops either an empathetic or repulsive response with theprobability of 𝑝 𝑖 or 1 − 𝑝 𝑖 , respectively. We assume that themagnitude of empathy and repulsion depends on the diﬀerence in opinion values.
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 introducefunction 𝑓 (𝑜𝑖 (𝑡 − ), 𝑜 𝑝𝑗 (𝑡 − )), which represents the change inthe opinion value, as
Δ 𝑐 𝑘 exp(−𝑘 |Δ |).

(1)= 𝑜 𝑝𝑗 (𝑡 − ) − 𝑜𝑖 (𝑡 − ) is the diﬀerence between

Note that Δ
the opinion value of the post seen by user 𝑖 in latest andthe user’s opinion value. Parameter 𝑘 (> 0) is a constant that
determines the decay in the intensity of empathy with respect
to the absolute value of the diﬀerence of opinion values |Δ |,and 𝑐 is a parameter that is adjusted to be 𝑜𝑖 (𝑡) ∈ [−1, 1],
which satisﬁes 𝑐𝑘 ≤ 1.
Next, consider the case of repulsion. In this case, itis necessary to develop a stronger repulsive reaction to theopinion values that are very diﬀerent from their own opinion
values, and the repulsion should be weaker when the opinion
values are close. To express the repulsive reaction in thesame form as in the case of empathy, we impose a periodic
boundary condition on the two ends of +1 and −1 to makethe diﬀerence between the opinion value of a post and theuser’s opinion |Δ ′ | = 2 − |Δ |; the smaller the diﬀerence |Δ ′ |,the larger the repulsive reaction. Based on this, we design
function 𝑓 (𝑜𝑖 (𝑡 − ), 𝑜 𝑝𝑗 (𝑡 − )), which represents the change inopinion value, as(1 − 𝑜𝑖 (𝑡 − )) 𝑐 𝑘 exp(−𝑘 |Δ ′ |) , (Δ ≤ 0),− (1 + 𝑜𝑖 (𝑡 − )) 𝑐 𝑘 exp(−𝑘 |Δ ′ |) , (Δ > 0).

(2)
(3)

Figure 1 illuminates the concrete actions of expressions(1), (2), and (3).
The expression (1) shows the change in the opinion
value in the case of an empathy reaction; it indicates that theuser’s opinion value approaches the opinion value of the postby an amount proportional to the diﬀerence, 𝑐 𝑘 exp(−𝑘 |Δ |).
In this case, the positive or negative value of Δ corresponds
to the 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. 1(a) and the positive direction of Fig. 1(b).
Figure 1(a) is the case of Δ ≤ 0. The arrow pointing in

-1

-0.5

0

0.5

1

0

0.5

(a) Δ ≤ 0

-1

-0.5

1

(b) Δ > 0

Fig. 1: Intuitive images of opinion value change.
the negative direction from the blue circle representing theuser’s opinion value indicates that the user’s opinion value
approaches the opinion value of a post by the proportion of𝑐 𝑘 exp(−𝑘 |Δ |) of the diﬀerence |Δ |. On the other hand,Fig. 1(b) is the case of Δ > 0, where the sign of Δ denotes a
positive change in opinion value in the opposite direction tothat of ﬁgure 1(a).
The expressions (2) and (3) show the change in opinion
value under a repulsion reaction. First, we will discuss thecase of Δ ≤ 0. As shown in Fig. 1(a), the opinion valueof user 𝑖 increases by the proportion 𝑐 𝑘 exp(−𝑘 |Δ ′ |) of thediﬀerence |1 − 𝑜𝑖 (𝑡 − )| between the opinion value of user 𝑖and the 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 Δ > 0. As shownin Fig. 1(b), the opinion of user 𝑖 decreases by the proportionof −(1 + 𝑜𝑖 (𝑡 − )) of the 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.
The parameter 𝑘 (> 0) represents the decay 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.
3.

Experimental evaluations of the proposed model

3.1

Preparation for evaluation experiments

We explain how to generate a sequence of posts and anindex of polarization for the experiments conducted on the
proposed model.
3.1.1

Generating a sequence of posts

Here, we generate artiﬁcial data using the multivariate
Hawkes process [5]. The multivariate Hawkes process isa point process in which multiple processes mutually excite

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)3

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
𝛼 𝑗𝑖 e−𝛽 𝑗𝑖 (𝑡−𝑡ℎ ) ,(4)𝜆𝑖 (𝑡) = 𝜇𝑖 +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 𝑡 ℎ is
its element. 𝛼 𝑗𝑖 and 𝛽 𝑗𝑖 ( 𝑗 = 1, . . . , 𝑁, 𝑖 = 1, . . . , 𝑁)represent the rate jump and decay rate for user 𝑖 due to user
𝑗’s posts, respectively. Thus, the posting rate of user 𝑖 isdetermined by the inﬂuence of the user-speciﬁc base rate
and the previous posts of neighboring users.
We generate a sequence of posts using the multivariate
Hawkes process. First, a social network with 10 nodes iscreated by the Barabási-Albert model (hereinafter referred
to as the BA model) [6]. In eq. (4), 𝛼 𝑗𝑖 and 𝛽 𝑗𝑖 are given
as uniform random numbers in the range of [0, 1] for eachcombination of 𝑗 and 𝑖 ( 𝑗, 𝑖 = 1, . . . , 𝑁). The base rate 𝜇𝑖
is given as a uniform random number in the range of [0, 1].
We evaluate the proposed model by generating a sequenceof posts in the observation period [𝑡1 , 𝑡 1 + 2] with the ﬁrst
post occurrence time 𝑡 1 .
3.1.2 Index of polarization
We introduce the index [7] for frequency distribution (𝝅, 𝒚) =(𝜋1 , . . . , 𝜋 𝑛 ; 𝑦1 , . . . , 𝑦𝑛 ) as the index of polarization:
𝑃=𝐾

𝑛 Õ
𝑛
Õ
𝑖=𝑖 𝑗=1

𝜋𝑖1+𝜃 𝜋 𝑗 𝑦𝑖 − 𝑦 𝑗 ,(5)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 [7], the stronger thesense a user has to 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
of polarization. For example, the maximum value is taken if
half of the users belong to the lowest and half to the highest
class, respectively, and the minimum value is taken if allusers belong to the same class.

(a) 𝑝 = 0.9, 𝑘 = 2(b) 𝑝 = 0.1, 𝑘 = 2(c) 𝑝 = 0.9, 𝑘 = 10

Fig. 2: Examples of opinion value change.

apply the opinion value change rule to the sequence of contributions generated in Sect. 3.1.1, with the probability ofan empathetic response and the probability of a repulsive
reaction being 𝑝 𝑖 = 𝑝 and 1 − 𝑝 𝑖 = 1 − 𝑝, respectively, for
all 𝑖.
Three typical simulation results with diﬀerent parameters for the opinion change using the proposed model are
shown in Figs. 2(a), 2(b) and 2(c). Parameter 𝑘 = 2 is ﬁxed
for the cases of Figs. 2(a) and 2(b), and the probability of
empathy 𝑝 is set to 𝑝 = 0.9 and 𝑝 = 0.1, respectively. The
diﬀerence in the probability of empathy reaction 𝑝 causes
a consensus of opinions in Fig. 2(a) and a polarization of
opinions in Fig. 2(b). Figure 2(c) shows the case of 𝑘 = 10.
Since the inﬂuence of other users’ posts is smaller than when𝑘 = 2, the opinion values of each user do not change muchfrom their initial values.
We then evaluate the polarization of the ﬁnal state at theﬁnal simulation time using the index of polarization (5), foreach combination of parameters 𝑝 and 𝑘. The normalization
 −(2+𝜃)Í𝑛parameters are set to 𝐾 =, the polarization
𝑖=1 𝜋𝑖sensitivity is set to 𝐾 = 0.5, and the number of classes is set
to 𝑛 = 10. Thus the possible values of the polarization index
are approximately [0, 0.62]. Assuming 𝑐 = 0.05, Fig. 3
shows the average of 10 simulations for each combination of
𝑝 = 0, 0.1, . . . , 1 and 𝑘 = 1, . . . , 10 with diﬀerent initial
opinion values. According to Fig. 3, the polarization at theend of the simulation can be divided into three main types:
concentrated into one opinion, polarized into two opinions,and little change in the opinions of each user. Opinions canbe concentrated into one opinion when roughly 𝑝 ≥ 0.4 and1 ≤ 𝑘 ≤ 5. This tendency occurred because they were more
likely to show an empathic reaction even when the diﬀerencein opinion values was only slight. The polarization into two
opinions occurs when 𝑝 < 0.4 and 1 ≤ 𝑘 ≤ 4. This tendency
occurred because they were more likely to show a repulsive
reaction even when the diﬀerence in opinion values was only
slight. The opinion of each user hardly changes when 𝑘 ≥ 5,
regardless of the value of 𝑝. As 𝑘 increases, it is due to thefact that we empathize only with those opinions that are very
close and reject only those that are far enough away.
4.

Comparison with previous research

3.2 Parameter characteristics of the proposed model
We conduct evaluation experiments to elucidate the characteristics of the parameters of the proposed model. The
experimental conditions are as follows. The initial opinion is given as a uniform random number of [−1, 1]. We

We compare our model with the opinion formation model,
which takes into account the reactions corresponding to empathy and repulsion. As an example, we take the BEBA
model [4]. The BEBA model is an expansion of classicalopinion formation model, DeGroot model [8]; it contains bi-

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)

p
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

4

0.5

0.4

0.3

0.2

(a) BEBA model(b) Proposed modelFig. 4: Change in opinion values when initial opinion values
given to all users have the same sign.

0.1

1

2

3

4

5

k

6

7

8

9

10

0.0

Fig. 3: Parameter characteristics of the proposed model.
ased assimilation and the backﬁre eﬀect. Biased assimilationmeans that we highly evaluate information that is consistent
with our original opinion, and the backﬁre eﬀect means thatwhen we are exposed to an idea that is diﬀerent from ourown, we come to strongly believe 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 the opinion values 𝑦𝑖 (𝑡) and𝑦 𝑗 (𝑡) are the same sign, biased assimilation controls, andif they are diﬀerent signs, backﬁre eﬀect dominates. Thismeans that the opinion value 0 is neutral and users have different opinions depending on whether they have positive or
negative opinion values. On the other hand, in the proposedmodel, the amount of change in the opinion value is determined based on the diﬀerence between the post’s and user’s
opinion values. To clarify the diﬀerence between the twomodels, we give all the initial opinion values with the same
sign in a comparative evaluation.
First, we describe the results of the experiments on the
BEBA model. Figure 4(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 backﬁreeﬀect was quiescent, and the opinions became concentrated
at the end of the simulation. Next, we describe the results ofthe experiments on the proposed model. Figure 4(b) shows
the change in opinion values for 𝑝 = 0.1 and 𝑘 = 2 yieldedby the proposed model. Although the initial opinion values
with the same sign were given to all nodes, the amount ofchange in the opinion values was determined based on thediﬀerence in the opinion values, so that a repulsion occurredand polarization commenced. Depending on the choice ofparameters, the proposed model was found to be able tohandle cases of polarization in which repulsive reactions
were active even with a biased opinion value distribution.
5. Conclusion
In this paper, we proposed a model of social media users’

opinion formation to elucidate the mechanism of opinion
polarization. The model is characterized by two types ofreactions: empathy and repulsion, with diﬀerent strengths
of inﬂuence being created by diﬀerences in user’s 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 with the concept of objective opinion neutrality, it has the advantage ofbeing able to respond ﬂexibly to a biased distribution of
opinion values. These features allow a wide range of user
characteristics to be appropriately modeled so as to wellcover the phenomenon of polarization. The dependence ofthe change in the opinion value on the initial values of theopinions will be examined in the future.
Acknowledgement
This research was supported by Grant-in-Aid for ScientiﬁcResearch 19H04096 and 20H04179 from the Japan Society
for the Promotion of Science (JSPS).
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