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PAPER

Special Section on Multidisciplinary and Innovative Key technologies in Advanced communication systems (MIKA)

A Spectral-Based Model for Describing Social Polarization in
Online Communities
Tomoya KINOSHITA†a) , Student Member and Masaki AIDA†b) , Fellow

SUMMARY The phenomenon known as social polarization, in which asocial group splits into two or more groups, can cause division of the society
by causing the radicalization of opinions and the spread of misinformation,is particularly significant in online communities. To develop technologies
to mitigate the effects of polarization in online social networks, it is necessary to understand the mechanism driving its occurrence. There are somemodels of social polarization in which network structure and users’ opinions
change, based on the quantified opinions held by the users of online social
networks. However, they are based on the interaction between users connected by online social networks. Current recommendation systems offer
information from unknown users who are deemed to have similar interests.
We can interpret this situation as being yielded non-local effects brought onby the network system, it is not based on local interactions between users. In
this paper, based on the spectral graph theory, which can describe non-local
effects in online social networks mathematically, we propose a model of polarization that user behavior and network structure change while influencing
each other including non-local effects. We investigate the characteristicsof the proposed model. Simultaneously, we propose an index to evaluate
the degree of network polarization quantitatively, which is needed for ourinvestigations.
key words: online social networks, polarization, Laplacian matrix, Fiedler
vector

1.

Introduction

In recent years, social media, including Twitter and YouTube,
have been spreading rapidly, and information exchange anddissemination on online social networks (OSNs) are being
actively promoted [1]. In particular, social networking services (SNS) have become indispensable in our daily lives. By
using them, we can efficiently access information matching
our interests and easily communicate with other users whoshare common interests [2]. Although the convenience of social media has been enhanced by providing individually optimized information to each user and promoting connections
among people through recommendation functions, these features have the potential for causing polarization among users[3]. In many studies, social polarization refers to the divisionof a social group into two groups. In this study, we consider
social polarization in a more generalized way by defining it
as division into two or more groups.
Many studies have addressed the occurrence of social
polarization in online communications on social media. For
Manuscript received November 8, 2021.
Manuscript revised February 14, 2022.
Manuscript publicized July 13, 2022.
† The authors are with Tokyo Metropolitan University, Hino-shi,
191-0065 Japan.
a) E-mail: poteti908@yahoo.co.jpb) E-mail: aida@tmu.ac.jp
DOI: 10.1587/transcom.2021MEP0001

example, [4] analyzed the Twitter discussions of several political groups and found that groups with different ideologies
interact less. This tendency was particularly noticeable between groups with opposing ideologies. In addition, [5]showed that individuals with similar ideologies exchanged
more information on political topics than on non-political
topics. [6] and [7] pointed out that the social polarizationof users may contribute to the spread of misinformation andfake news in social networks. To develop technologies tomitigate the effects of polarization, understanding the mechanism driving its occurrence is necessary.
Some of the human factors that may contribute to theoccurrence of polarization include confirmation bias [8] (focusing only on information that conforms to self view) andthe backfire effect [9] (reinforcing own opinions when exposed to opposing opinions). Some of the characteristics ofsocial media include filter bubbles [10] (the preferential presentation of information of interest to each user) and echochambers [11] (the reinforcement of user beliefs throughrepeated interactions within a closed community). There
are many reports that these phenomena have been observedin discussions of political topics in SNS [12], [13]. By
proposing a model that describes them, we believe that it ispossible to establish a basis for considering universal countermeasures that can attenuate polarization in various kindsof situations.
Related work has proposed some models of polarization that posit that the quantified opinions held by users
in social networks influence network structure changes andthe dynamics of user opinion formation. [14] introduced
a mathematical model to describe online discussions andthe attendant polarization; they concluded that the repulsionof opinions among users can explain its occurrence. [15]proposed a model in which user opinions and social interactions change with the viewing of social media posts madeby users. They showed that segregated communities emerge
through social media mechanisms. These models describe
the process by which the opinions of each user affect the
network structure. However, network structure changes do
not directly affect the user opinion formation. Also, thoughsome models of polarization have been proposed in which
the opinions of users and connections between users mutually interact [16], [17], each user is only directly affectedby the users with whom it has some connection. Therefore, they cannot describe the non-local effects yielded by
the recommendation functions of network systems. Somemodels have also been proposed to investigate the effect of

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

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recommendation functions on polarization [18], [19]. However, even in those frameworks, the opinion change of eachuser is not influenced by unknown users. In addition, [19]focused on the function of recommending items, which isoften used in online shopping sites. Since the relationshipbetween users and items is different from the relationshipbetween users, however, its framework cannot describe thefunction of recommending users in OSNs.
In general, user interest and network structure should
be related, and therefore they should interact. When theopinions and interests of each user change, the relationships
among users is also expected change as a result. In addition,
users can view information transmitted by unknown users
in OSNs and can receive information from unknown users
who are deemed to have similar interests, so they may be
influenced by unknown users. These effects are non-local
interactions not based on the communication through links in
OSNs. Based on these considerations, we propose a model ofuser polarization that takes account of OSN structure. In our
proposed model, we treat OSN structure in the frameworkof spectral graph theory. In particular, we focus on the
Fiedler vector, which is a kind of eigenvector of the Laplacianmatrix, for describing non-local effects. It enables us todescribe the process by which user opinions and network
structure change while influencing each other, as well as tomodel user behaviors such as following and unfollowing,that are common in social networks. Our proposed model
also takes into account the unique characteristics of eachuser, such as leader and follower, so that it is a frameworkthat takes a realistic view of OSNs. Our model is effectivein understanding the non-local effects among users in OSNs
and the impact of the interaction between opinion change
and network structure change on polarization.
To evaluate the proposed model properly, we need aquantitative index to measure the degree of polarization.
Many studies that propose indices of polarization focus on
the division of user opinions or network structure into twogroups [20], [21]. However, few indices allow for division
into three or more groups. In this study, we propose an index
to evaluate various types of polarization.
The rest of this paper is organized as follows. Section 2
explains spectral graph theory and the concept of the modelof polarization proposed in this study. Section 3 proposes
an index to quantitatively evaluate the degree of network polarization. Section 4 conducts some experiments to evaluate
the model and clarify its parameters’ characteristics. Finally,
we state conclusions and future work in Sect. 5.
2.

2.1

Fiedler Vector and Similarity of User Opinions

Let G(V, E) be an (unweighted) undirected graph representing the structure of an OSN with N nodes, where V is the setof nodes, and E is the set of undirected links. Here, nodes
and links represent users and the relationships between them,
respectively. When there is a link between nodes i and j,they can communicate directly. The (unweighted) adjacency
matrix A := [Ai j ]1≤i, j ≤ N is an N × N matrix defined as(
1, (i, j) ∈ E,
Ai j :=(1)0, (i, j) < E.
If node i has degree di , the degree matrix D is an N × N
matrix defined as
D := diag(d1, d2, . . . , d N ).

(2)

The Laplacian matrix L of G(V, E) is defined as
L := D − A.

(3)

It is known that Laplacian matrix L has a minimum eigenvalue of zero. The multiplicity of eigenvalue zero is equal
to the number of connected components of G(V, E).
Assume that G(V, E) is a connected undirected graph.
Let us sort the eigenvalues of the Laplacian matrix of G(V, E)
in ascending order as follows:
0 = λ0 < λ1 ≤ · · · ≤ λ N −1 .

(4)

The smallest eigenvalue that is not zero (the second smallest
eigenvalue λ1 ) is called the algebraic connectivity, and it isused as a measure of how tightly connected the undirected
graph is. Eigenvector v1 of L associated with λ1 is known
as the Fiedler vector [24].
Figure 1 shows an unweighted undirected graph withfour densely connected subgraphs connected by some links,
and Fig. 2 shows the components of the Fiedler vector of the
Laplacian matrix of the graph shown in Fig. 1. From thesefigures, we can see that the four cluster structures of the plotted points correspond to the four subgraph structures which
are characteristic of the graph. This example demonstrates

A Spectral-Based Model of Polarization

In this section, we describe a spectral-based model of social
polarization. First, based on [22], [23], we introduce theLaplacian matrix to express network structure. Then, we
describe the concept of the model that reflects user behavior
and characteristics in OSNs.
Fig. 1 Unweighted undirected graph with four densely connected subgraphs connected by some links.

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Fig. 3Fig. 2

why the Fiedler vector can be taken as a useful index forunderstanding graph structure.
Based on the above, we can connect OSN structure
and users’ opinions or tastes. Components of the Fiedler
vector that have similar values can be interpreted as similaruser opinions or tastes. Accordingly, we propose a model ofpolarization that determines the opinion or taste of each node
based on the value of each component of the Fiedler vector
and change the network structure based on them. Here, notethat the values of the components are not directly related
to the strength of opinions. The fact that the values of thecomponents of the Fiedler vector are close indicates that thecorresponding users have similar opinions and tastes. Thisidea makes it possible to apply the model to not only topics
where opinions fall into two groups, which has been thetarget of related work, but also topics where more diverse
opinions are likely to be formed. We will describe how todetermine the opinion value of each node based on eachcomponent of the Fiedler vector, along with the concept ofthe model of polarization.
2.2

Example of leader and follower nodes.

Components of the Fiedler vector corresponding to Fig. 1.

Concept of the Model of Polarization

In social media, including SNS, each user can freely follow
or unfollow other users as a response to the posts displayed on
the screen. Also, we think that some users in social networks
influence their surroundings by acting on their own opinions
and beliefs, while others form their own opinions and build
human relationships through the influence of other users.
Our model of polarization reflects these user behaviors andcharacteristics.
To reflect the user characteristics of OSNs in our model,
we introduce two types of characteristic nodes on the undirected graph representing the social network. One is theleader node; it represents users who act with consistent opinions and beliefs. This node has the constraint that the opinion
value of the node will remain fixed at the component of theFiedler vector in the initial state, regardless of the opinionchanges caused by network structure changes. The otheris the follower node; it represents users who form opinions
and human relationships as influenced by their surroundings.

This node has the constraint that it will never disconnect its
link with the leader nodes. We call the node that is neither leader nor follower the other node. Here, we regard theopinion values of the follower node and the other node as thecomponents of the Fiedler vector. That is, they will changewith the network structure. Figure 3 shows an example ofleader and follower nodes. The red links in the figure will
never be disconnected even under network structure changes.
Here, we assume that the characteristic of each node does
not change regardless of network structure changes.
Next, we explain the procedures for disconnecting andconnecting links to describe the changes in the relationships
between users. At each time step, we select a node at random
and find its adjacent node whose opinion value is the most
distant from that of the selected node. Then, we disconnect
the link between them only if both of the following two
conditions are satisfied:• The link does not connect a leader node to a follower
node.
• The graph remains connected if we disconnect the link.
The process of disconnecting links recreates the fade outof relationships between users with different opinions andtastes.
Then, if we disconnected the link, we select one nodethat is not adjacent to the first selected node and connect
a new link between them. Here, we provide two ways of
selecting the node:(i) With probability φ, we randomly select one node fromthe set of nodes that are not adjacent to the first selectednode.
(ii) With probability 1 − φ, we select one node that is not
adjacent to the first selected node and that has the closest
opinion value to it.
Since (i) is a link connection process that does not dependon the opinion value of each node, we can assume that itcorresponds to a coincidental encounter in social networks.
On the other hand, (ii) expresses the process of connecting
people with similar opinions and tastes, so we can assume
that it corresponds to non-local effects such as the encounterscreated by the recommendation functions in social media.

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In the environment of OSNs, random link connections asin (i) are rare. Therefore, we believe that setting φ to asmall value can realistically represent the process of building
relationships between users in OSNs.
According to the rules described above, OSN structure and the opinion value of each node will change while
influencing each other by repeating the process of link disconnection and connection. This model does not change
the total number of links, but the degree of each node canchange.
3.

Index of Social Polarization

In Sect. 2, we proposed a model of polarization for changing
the network structure based on the opinion value of eachnode. However, to determine whether the network after astructure change is polarized or not, a quantitative measure isnecessary. Clustering coefficients are attractive as indices toevaluate the strength of polarization. This is because sparseand dense parts are visible in a polarized network structure,and the corresponding clustering coefficient is expected tobe high. However, it is difficult to distinguish between dense
networks (e.g., complete graphs) from polarized networks
using clustering coefficients because those graphs also havehigh values. Moreover, they cannot determine how manygroups exist in the network. For these reasons, we believethat clustering coefficients cannot be used to quantify polarization. Therefore, in this section, we propose an index
of polarization that describes both the number of polarizedgroups and the strength of the polarization effects in the
network based on the distribution of the opinion value ofeach node. The procedure to derive the index includes two
steps: determination of the number of polarized groups, andevaluation of the strength of the polarization effects.
We start by deleting the nodes with degree 1 from the
network. This operation corresponds to deleting terminal
nodes in network analysis when extracting the network structure of an SNS. This process prevents us from being unable
to capture the rough characteristics of the network structure
due to the influence of small degree nodes.
Next, we normalize the opinion value of each node.
This process enables us to handle the component distributionof the Fiedler vector, which greatly depends on the numberof nodes in the network and its structure, in a unified manner.
Let oi be the opinion value of node i, omin be the minimum
opinion value among all nodes, and omax be the maximum
value. We define the normalized opinion value oi0 of node i
as follows:oi0 =2 (oi − omin )− 1.
omax − ominvalues evenly into n intervals, and obtain the frequency distribution by counting up the number of opinion values ineach interval. If the frequency in an interval is greater thanor equal to a threshold θ, then we consider the nodes with
opinion values in the interval to be one group. That is,threshold θ represents the minimum number of nodes thatcan be considered as a group. In addition, threshold θ restricts the maximum number of polarized clusters. Here, we
introduce some conditions that should be satisfied between
the scale of the network and the number of polarized groups
contained in it:• As the scale of the network grows, the number of polarized groups that can be formed increases.
• A group with a very small scale relative to the scale ofthe entire network is not considered a polarized group.
By considering the above two conditions, we define threshold
θ as
h√ i
θ= N ,(6)
where N is the number of nodes in the network and [x] is thelargest integer less than or equal to x. Based on the above,let mn be the number of polarized groups in the network andgn be the mn -dimensional vector with the number of nodes
in each polarized group as each component, where n is thenumber of intervals. Note that the range constraint on n,
described below, guarantees mn ≥ 1.
As an example, we obtain mn and gn from the components of the Fiedler vector shown in Fig. 2, that is, theopinion value of each node. Figure 4 shows the opinion
values normalized to the range [−1, +1] using (5). Table 1
shows the frequency distribution table when the number ofintervals, n, is four. Threshold θ = 4, since the number ofnodes N = 22. From Table 1, since the frequencies of allintervals are greater than or equal to θ, we obtain m4 = 4 andg4 = (6, 5, 5, 6).
Using mn and gn , we define index P(n) for the degreeof polarization of the network structure as follows:(5)

The minimum and maximum values of the normalized opinion value are −1 and +1, respectively.
Next, we evaluate the number of polarized groups in
the network and the number of nodes in each polarizedgroup based on the distribution of normalized opinion values. We divide the range [−1, +1] of normalized opinion

Fig. 4

Normalized opinion values.

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Table 1Frequency distribution table (n = 4).
Opinion value−1.0∼−0.5−0.5∼0.0
0.0∼0.5
0.5∼1.0

Table 2
n
3
4
5

mn
2
4
4

Frequency
6
5
5
6

4.

Example of calculating each index.
gn(10, 10)(6, 5, 5, 6)(6, 5, 5, 6)

P1 (n)
0.909
1.00
1.00

P2 (n)
2.00
1.00
1.00


mn = 1, 0,
P1 (n)
P(n) :=, mn ≥ 2.
p P2 (n) + 1

P(n)
0.525
0.707
0.707

(7)

Here, P1 (n) and P2 (n) in (7) are defined asmn
1 Õgn (i),
N i=1
2
mn 
Õ
N
− gn (i) ,P2 (n) :=mn
i=1P1 (n) :=(8)(9)

where gn (i) is the i-th component of vector gn . P1 (n) is anindex that indicates how many nodes belong to some polarized group out of all the nodes in the network. The larger thevalue of P1 (n) is, the greater the degree of polarization is.
P2 (n) is an index that indicates how unbalanced the numberof nodes in each group is. The smaller the value of P2 (n) is,
the greater the degree of polarization is. P(n) is determinedby P1 (n) and P2 (n), and the range of its value is [0, 1], where
the larger the value is, the more strongly polarized
p the network structure is. In addition, since P1 (n) and P2 (n) + 1
have the same dimension, index P(n) is a dimension-lessmeasure. The reason why P(n) = 0 when mn = 1 is because
we define social polarization as division into two or moregroups, as described in Sect. 1.
Finally, we calculate P(n) for each integer n satisfying
3 ≤ n ≤ [N/θ] and define the index P∗ of polarization as
follows:P∗ := max P(n).
n(10)

Based on the above, we quantitatively determine the degreeof polarization from P∗ and the number of polarized groups
on the OSN from mn used to calculate P∗ .
Table 2 shows examples of calculating each index fromthe distribution of opinion values shown in Fig. 4 in the range
3 ≤ n ≤ [N/θ] = 5. Note that P1 (n), P2 (n), P(n) values are
written to three significant digits. From Table 2 and Eq. (10),the degree of polarization of the network structure shown in
Fig. 1 is calculated as P∗ = 0.707. Considering this result
together with the fact that the number of polarized groups
when P(n) is maximum is mn = 4, we believe that the valuesof P∗ and mn appropriately reflect the characteristics of the
network structure shown in Fig. 1.

Basic Characteristics of the Proposed Social Polarization ModelIn this section, we examine through simulations how themodel parameters and the characteristics of nodes introducedin the model described in Sect. 2 affect the social polarizationof the network structure.
The experiments use unweighted undirected graphswith the number of nodes N = 102 generated by the ErdösRényi model (ER model) [25] and the Barabási-Albert model(BA model) [26] as network models. Here, the link generation probability in the ER model is set to 0.05, and thenumber of nodes in the initial state in the BA model is set tothree. Figures 5 and 6 illustrate the corresponding network
structures. The red nodes represent leader nodes, the blue
nodes represent follower nodes, and the yellow nodes represent the other nodes. The number of leader and follower
nodes in Fig. 5 is 2 and 19, respectively, and in Fig. 6 it is
2 and 44, respectively. In the upper panels of Figs. 5 and6, leaders are the nodes with the largest or smallest opinion
values, and followers are selected randomly. On the other
hand, in the lower panels of Figs. 5 and 6, leaders are thenodes with the first or second highest degree, and followers
are the nodes adjacent to the leader. We apply the model toeach network and calculate the indices P∗ and mn from the
network after structure changes. As an experimental condition, the simulation time for the model, t, was set to 3000.
Also, from (6), the threshold used to calculate the indices
was set at θ = 10.
Figures 7 and 8 show index, P∗ , of polarization andthe number, mn , of polarized groups for various φ values,
using the networks shown in the upper and lower panels of
Fig. 5 as the initial conditions, respectively. We conducted
the simulation 100 times for each value of φ by changing theseed used in random node selection at each time step, so P∗
and mn in the figures are averages. Note that the error bars
represent the 95% confidence interval.
Focusing on the averages of P∗ in Figs. 7 and 8, we cansee that P∗ takes larger values as parameter φ decreases. As
mentioned in Sect. 2, a smaller value of φ means that users
with similar opinions are more likely to connect as a resultof the network structure change. Also, we design the modelof polarization so that the links between nodes with different
opinions and interests are disconnected. Therefore, theseresults indicate that in the random network generated by the
ER model, polarization is more likely to occur due to these
two factors, regardless of how the leader and follower nodes
are defined.
Next, focusing on the averages of mn in Figs. 7 and 8,
we can see that decreasing φ increases the value of mn . This
indicates that the tendency for users with similar opinions
to connect contributes to forming many polarized groups in
the network. Here, for φ = 0 to 0.3, the averages of mn are
greater than two, even though the number of leader nodes
is two. It means that groups without leader nodes are likely
to appear in the network. We believe that the reason is the

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Fig. 5102).

Unweighted undirected graph generated by the ER model (N =degree of the leader nodes in their initial state. When theleaders are the nodes with the largest or smallest opinion
values, the smaller φ is, the more difficult it is for them toacquire more links because their opinion values are far fromthose of the other nodes. Also, when the leaders are thenodes with large degree, their degree is at most one-tenth of
the total number of nodes in the network. Even if a group
including the leader is formed, its scale is not expected to bevery large. Therefore, there are many nodes that do not forma group with a leader, and the number of polarized groups
without a leader will increase.
To further understand the influence of the characteristics of the nodes introduced in our model, we also simulated the situation in which leaders change their opinions(Appendix). The simulation results indicate that whether theleaders change their opinions or not is not an important factor
in polarization. Thus, we reiterate our assumption that theleader holds a consistent opinion. In the experiments using
the network generated by the BA model, we do so under thecondition that leaders keep their opinions unchanged.
Figures 9 and 10 show index P∗ of polarization and thenumber mn of polarized groups for various φ values, using
the networks shown in the upper and lower panels of Fig. 6 as

Fig. 6102).

Unweighted undirected graph generated by the BA model (N =initial conditions, respectively. The number of simulations
and the meaning of the error bars are the same as thosementioned above.
Figure 9 indicates that decreasing φ increases P∗ andmn . It is similar to the results of the network generated
by the ER model. Therefore, under this initial state, we
can confirm that the tendency to connect users with similar
opinions also contributes to polarization in the scale-free
network generated by the BA model.
However, focusing on the averages of P∗ in Fig. 10, we
can see that the value of P∗ is small regardless of the value ofφ. This result is very different from the characteristic mentioned above. To discuss the reason for this, let us consider
the networks in the initial state. In the network generated by
the BA model, nodes with a large degree tend to connect, so
the opinion values of the leader nodes are close to each other.
In addition, the followers determined as the adjacent nodesof the leaders have the constraint of keeping their links withthe leaders, so their opinion values tend to approach that of
the leader. Therefore, we believe that index P∗ of polarization is small due to the formation of a large-scale group that
includes two leader nodes and many follower nodes.
Focusing on the averages of mn in Fig. 10, we can see

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Fig. 7 Average of P ∗ and mn for parameter φ using the upper panel ofFig. 5 as the initial network.

Fig. 9 Average of P ∗ and mn for parameter φ using the upper panel ofFig. 6 as the initial network.

Fig. 8 Average of P ∗ and mn for parameter φ using the lower panel ofFig. 5 as the initial network.

Fig. 10 Average of P ∗ and mn for parameter φ using the lower panel ofFig. 6 as the initial network.

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Fig. 11 Network and the components of Fiedler vector in ascending order(t = 3000, φ = 0, P ∗ = 0.176).

that decreasing φ increases mn . However, comparing the
averages of mn shown in Figs. 7–10, we find that those of
Fig. 10 are the smallest overall among those of the others.
We think that this is due to the existence of the large-scalegroup discussed in the previous paragraph. The formationof a large-scale group means that the number of nodes notincluded in it tends to be small. Therefore, it is hard for those
nodes to form a new group.
Here, we describe further experimental results, focusing on the scale-free network generated by the BA model.
To facilitate understanding of the relation among index P∗of polarization, the network topology, and the distributionof opinion values, we show some examples of them after
structure change.
Figure 11 shows the network and the components of
the Fiedler vector in ascending order at t = 3000 withP∗ = 0.176. For the experimental conditions, we used the
network shown in the upper panel of Fig. 6 as the initial condition with parameter φ = 0. These figures indicate that there
are multiple community structures in the network and thatthe components of the Fiedler vector distribute in a staircase
pattern with several peaks. This result shows that a polarized
structure appears in the network even if P∗ is around 0.15,
although it is far from the complete polarization representedby P∗ = 1. It is also notable that leader nodes do not belong
to any community in the network. The reason may be that itis hard for the leaders to acquire links because random linkconnections cannot occur under this experimental condition.

Fig. 12 Network and the components of Fiedler vector in ascending order(t = 3000, φ = 1, P ∗ = 0.025).

Nevertheless, such a polarized situation occurs regardlessof leaders because each node connects with nodes that have
similar opinions and blocks nodes that have different opinions.
Figure 12 shows the network and the components of
the Fiedler vector in ascending order at t = 3000 withP∗ = 0.025. For the experimental conditions, we used the
network shown in the upper panel of Fig. 6 as the initial condition with parameter φ = 1. Under this condition, there isno community structure in the network, and multiple peaks
do not appear in the distribution of the components of theFiedler vector. In this case, even if it is determined by thecalculation of the index that there are several groups in the
network, the number of nodes among them is highly unbalanced. Therefore, the value of P∗ is smaller than that ofFig. 11.
Finally, to understand how polarization strengthchanges with time, we investigate the time evolution of polarization index P∗ and the number of polarized groups, mn .
Figure 13 shows the simple moving averages of the time evolution of P∗ and mn for parameter φ using the upper panelof Fig. 6 as the initial condition. From these figures, we cansee that P∗ and mn decrease regardless of the value of φ at
the beginning, but then increase as the value of φ decreases.
That is, we can assume that the process of polarization begins with the formation of a large-scale group, which thengradually divides into multiple groups. Also, under φ = 0,

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Fig. 13 Time evolution of P ∗ and mn for parameter φ using the upper
panel of Fig. 6 as the initial condition (simple moving average of interval
width 19).

Fig. 14 Time evolution of P ∗ and mn for parameter φ using the lower
panel of Fig. 6 as the initial condition (simple moving average of interval
width 19).

where there is no randomness in the link connection, thevalues of P∗ and mn are stable in the latter half of the simulation time. On the other hand, under φ = 0.1 to 0.3, where
random link connections occasionally occur, they seem tohave increasing tendencies even at t = 3000. These results
indicate that random link connections may have the effect ofslowing or stopping the progression of polarization.
Figure 14 shows the simple moving averages of thetime evolution of P∗ and mn for parameter φ using the lower
panel of Fig. 6 as the initial condition. Focusing on thetime evolution of P∗ , it hardly increases after decreasing
regardless of the value of φ. The reason seems that theconstraint that the link between a leader and a follower isnever disconnected prevents the division of the large-scalegroup. However, since the other nodes do not have such a
constraint, they can form small groups. It may be related to
the result that mn increases with time when φ is small.

tenuation of relationships between users who have different
opinions. We also found that when the hubs in the scale-free
network are leaders and the users around them are followers,a large-scale group including them can readily form as the
network structure changes. Meanwhile, we found that increasing the number of interactions that are independent ofuser opinion values mitigated the polarization regardless of
the initial network structure and how leaders and followers
were determined, which is seen as an important finding fordeveloping countermeasures to polarization.
In future work, we will examine the impact of the initial
network structure on polarization in more detail. We will
also consider extending the applicability of the model sothat it can handle weighted undirected graphs and directed
graphs.

5.

This research was supported by Grant-in-Aid for Scientific
Research 19H04096, 20H04179 and 21H03432 from theJapan Society for the Promotion of Science (JSPS), and TMU
local 5G research support.

Conclusion

In this paper, we use the components of the Fiedler vectorin spectral graph theory to propose a model of social polarization in which user behavior and network structure change
while influencing each other. We also proposed an index thatcan evaluate social polarization quantitatively. Our model
reflects the non-local effects of user behavior, and the characteristics of users on OSNs. Simulations revealed that social polarization is more likely to occur due to the synergistic
effect of connecting users who have similar opinions with at-

Acknowledgments

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Appendix:

Experimental Results in the Case Where theLeaders’ Opinions are Variable

In the main text, we assumed that leaders have consistent
opinions and that the opinion values of the leader nodes
remain unchanged regardless of network structure changes.
However, in some cases, leaders may pander to the opinionsof their followers and change their own opinions. In thissection, we show some simulation results when the opinion
values of all nodes, including leader nodes, are regarded ascomponents of the Fiedler vector.
Figure A· 1 shows the results when the upper panelof Fig. 5 is the initial condition and the leaders’ opinions
are variable. Compared to the results when the leader’sopinion is invariant (Fig. 7), there are slight differences in
the averages of index P∗ of polarization and the number
mn of polarized groups, but the tendency that they increase
as φ decreases remains. These results imply that whether
networks and opinions become polarized or not is mostly
independent of whether leaders change their opinions or not.

Fig. A· 1 Average of P ∗ and mn for parameter φ in the case where theleaders’ opinions are variable (initial network: the upper panel of Fig. 5).

KINOSHITA and AIDA: A SPECTRAL-BASED MODEL FOR DESCRIBING SOCIAL POLARIZATION IN ONLINE COMMUNITIES

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Tomoya Kinoshita
received the B.E. degree
in Engineering and the M.E. degree in Computer Science from Tokyo Metropolitan University, Japan, in 2020 and 2022, respectively. Currently, he works for KDDI Corporation. He isinterested in the analysis of social network dynamics.

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 andORSJ.