IEICE TRANS. COMMUN., VOL.E106–B, NO.5 MAY 2023

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PAPER

Cluster Structure of Online Users Generated from Interaction
Between Fake News and Corrections
Masaki AIDA†a) , Fellow, Takumi SAKIYAMA†b) , Ayako HASHIZUME††c) , Nonmembers,
and Chisa TAKANO†††d) , Member

SUMMARY The problem caused by fake news continues to worsen
in today’s online social networks. Intuitively, it seems effective to issue
corrections as a countermeasure. However, corrections can, ironically,strengthen attention to fake news, which worsens the situation. This paper
proposes a model for describing the interaction between fake news andthe corrections as a reaction-diffusion system; this yields the mechanism
by which corrections increase attention to fake news. In this model, theemergence of groups of users who believe in fake news is understood as
a Turing pattern that appears in the activator-inhibitor model. Numericalcalculations show that even if the network structure has no spatial bias, theinteraction between fake news and the corrections creates groups that arestrongly interested in discussing fake news. Also, we propose and evaluatea basic strategy to counter fake news.
key words: fake news, reaction-diffusion system, activator-inhibitor system,
Turing pattern

1.

Introduction

Social networking service (SNS) has deeply penetrated into
our social life, and online user dynamics now greatly influence user behavior in both online societies and the real world.
In particular, fake news has had a great impact on the realworld, and it is said that 70% of claims about Trump during
the 2016 US presidential election were “false” or “mostly
false” information [1]. Since it is impossible to eradicate
fake news from the world, it is necessary to build a mechanism that is less susceptible to the adverse effects of fakenews on the premise that modern society will have to coexist
with fake news. For that purpose, it is necessary to understand the mechanism by which fake news spreads acrossonline social networks (OSNs) and to establish an effective
countermeasure.
1.1

Problem Description

The current solution to fake news is to proactively promote
Manuscript received April 4, 2022.
Manuscript revised July 6, 2022.
Manuscript publicized November 21, 2022.
† The authors are with Tokyo Metropolitan University, Hino-shi,
191-0065 Japan.
†† The author is with Hosei University, Machida-shi, 194-0298
Japan.
††† The author is with Hiroshima City University, Hiroshimas-shi,
731-3194 Japan.
a) E-mail: aida@tmu.ac.jpb) E-mail: sakiyama-takumi@ed.tmu.ac.jpc) E-mail: hashiaya@hosei.ac.jpd) E-mail: takano@hiroshima-cu.ac.jp
DOI: 10.1587/transcom.2022EBP3059

the correct information that denies the content of fake news.
Certainly, this seems to be intuitively valid. However, in
reality, the results are not so direct. A case has been reported
in Japan where the information issued to correct fake news
did not work properly [2]–[4].
The details of the matter are as follows. First, in late
February 2020, at the beginning of the COVID-19 epidemic,
a claim that toilet paper was out of stock was posted on
SNS. The claim was that shortages of toilet paper were imminent as the production area of toilet paper, in China, wasnegatively impacted by COVID-19. In reality, there was sufficient inventory and production capacity in Japan, so stock
shortages would never occur with normal demand. For thisreason, this post was non-factual fake news. At this point,
the fake news itself did not receive much attention and did
not trigger the hoarding of toilet paper. To counter this fakenews, corrections were disseminated basically saying that
toilet paper was not in short supply and that the originalreport was fake news. Many users who received this correction predicted that many other people would try to hoard
toilet paper due to the fake report, which stimulated a senseof crisis. In addition, the spread of the corrections attracted
the attention of users to the fake report, and as a result, thespread of fake news was actually promoted. As a result, thedissemination of the corrections to the fake report stimulated
the hoarding of toilet paper, ironically resulting in a shortageof toilet paper.
This case reinforced the difficulty of countering fakenews in that the dissemination of correct information does
not always lead to good results. In particular, since corrections can activate fake news, it has been suggested that it isnecessary to consider the interaction between fake news andcorrections in order to properly evaluate the effect of any
correction being considered. In [2], this case is analyzed by
the data-scientific approach and the authors concluded thatthe corrective information promoted unnecessary purchasing
behavior. On the other hand, this paper aims to understandthe mechanism of generating user clusters that discuss fakenews actively from the viewpoint of a theoretical model.
They are summarized in Table 1. Most simple positionis (a) in Table 1. It describes the process by which a user
who was initially unaware of fake news becomes aware of
fake news through the propagation of information. In thisposition, the spread of fake news is often described by using
the analogy of infectious disease models [5]–[7]. However,
the toilet paper case mentioned above was not caused by

Copyright © 2023 The Institute of Electronics, Information and Communication Engineers

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Table 1

Basic position of this research.

(a) Do users know fake news?
(b) Assuming that users knowboth fake news and corrections(c) Assuming that users knowboth fake news and corrections

YES
Believe infake news
User behavior is
influenced by bothNO
Believe incorrection
User behavior is
NOT affected by either

knowing or not knowing fake news. It is the position of(b) to deal with the conflict of different information. Whenboth fake news and its corrections propagate on OSNs, theuser is in the position of which information to believe andact. In this position, the spread of different information isoften described by competing contagion processes [8]–[10].
However, in the toilet paper example, the hoarding users did
not believe the fake news. The hoarders tried to buy toilet
paper before stocks were exhausted by other users, eventhough they knew the information was fake news. From thispoint of view, it is not important which information the users
believe in. It is the position of (c) to consider whether theuser’s behavior is influenced by fake news, and this is the
viewpoint of this paper.
This paper proposes a model describing the diffusion of
fake news on a social network using the activator-inhibitor
approach of reaction-diffusion systems. Also, based on the
proposed model, this paper demonstrates that the interaction
between fake news and corrections triggers the clusteringof nodes with a high interest in fake news. In addition,
this paper discusses a sufficient condition that prevents thegeneration of these user clusters.
1.2

Research Objectives

Many models have been proposed to describe the process bywhich fake news spreads over OSNs. They can be classified
into models based on the user’s own characteristics [11],[12] and models based on the environment surrounding theuser [13]–[16]. However, since these models deal with the
spread of one type of information, they fail to describe theinteraction between fake news and corrections. Therefore,
it is necessary to consider the characteristics of fake news
and corrections and develop an information spreading modelthat considers their interaction.
This paper is an extended version of [18]. In this paper, we consider the individual characteristics of fake news
and corrections and their interaction and propose a modelin which the two types of information are intertwined andspread across OSNs. The proposed model is based on an
activator-inhibitor model of the reaction-diffusion system.
This model shows that even if the network structure itself
is not initially biased at all, the interaction between fakenews and the corrections may result in the formation of OSN
clusters that believe in fake news. The evaluation in [18]assumed the state in which fake news and corrections wererandomly distributed. In this paper, we evaluate the situationin which fake news is already widely distributed but not seen
as particularly well noticed before the correction spread from
specific nodes. This follows the actual toilet paper shortage

case. The result can explain the mechanism whereby thecorrections amplify the influence of fake news.
Also, in addition to [18], we consider the sufficientcondition to prevent the amplification of the influence of
fake news and evaluate a basic strategy to counter fake news.
2.

Preliminary

2.1

Reaction-Diffusion System

Reaction-diffusion systems are mathematical models that describe phenomena in which multiple objects are distributed
in a certain space and diffuse in the space while interactingwith each other. The models can be written as reactiondiffusion equations. For example, consider two types ofobjects in a two-dimensional Euclidean space whose coordinates are written as (x, y). Let the density functions representing the spatial distribution of objects a and b at time t bea(x, y; t) and h(x, y; t), respectively. The reaction-diffusion
equations are given as follows:∂a(x, y; t)= f (a, h; t) + κa 4a(x, y; t),∂t∂h(x, y; t)= g(a, h; t) + κh 4h(x, y; t),∂t(1)where, f (a, h; t) and g(a, h; t) in the first terms on the righthand sides are called reaction terms. Since the reaction
terms are functions of a(x, y; t) and h(x, y; t), they describe
the effect of the interaction between the two types of objects.
In particular, because we consider local interactions that are
locally determined at each point (x, y), the reaction terms are
determined by the information at the point (x, y) given byf (a, h; t) = f (a(x, y), h(x, y); t),g(a, h; t) = g(a(x, y), h(x, y); t).

(2)

The second terms on the right-hand side are called the
diffusion term, they describe how the two types of objects
diffuse spatially. 4 denotes the Laplacian
4=

∂2
∂2
+
,
∂ x2 ∂ y2and κa and κh are diffusion coefficients that describe thediffusion intensity of each object. The larger the diffusion
coefficient, the faster is the object diffuses.
The reaction-diffusion system is widely applied in not
only the field of chemistry but also other fields such as biology, ecology, and physics.
2.2

Activator-Inhibitor Model

The activator-inhibitor model, which is a type of reactiondiffusion system, has been applied in network engineering [19], [20]. The application examples utilize the Turing
pattern that is characteristic of the activator-inhibitor model.
The Turing pattern is a kind of spatial pattern that is spontaneously generated and is known to explain the generation

IEICE TRANS. COMMUN., VOL.E106–B, NO.5 MAY 2023

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Fig. 1

Basic mechanism of the activator-inhibitor system.
Fig. 2of the geometric patterns that can appear on the surfaces ofanimal bodies.
As a simple and natural way of giving the reaction
terms, we can use the following linear model:f (a, h; t) := ca a(x, y; t) − sa h(x, y; t)− µa a(x, y; t) + ρa,g(a, h; t) := ch a(x, y; t) − µh h(x, y; t) + ρh ,

(3)
(4)

where ca , sa , µa , ρa , ch , µh and ρh are positive constants.
In the activator-inhibitor model, a(x, y; t) is called theactivator and h(x, y; t) is called the inhibitor. The intent of
this naming can be understood by looking at the structureof the reaction terms in (3) and (4). The reaction terms canbe explained qualitatively as follows. The presence of theactivator acts to promote the increase of both the activatorand the inhibitor at that location, while the presence of theinhibitor acts to suppress the increase of the activator atthat location. The first terms on the right hand sides of (3)and (4), and the second term on the right hand sides of (3)represent these effects. The third term on the right-handsides of (3) and the second terms on the right-hand sides of(4) represent the reaction of self-decay, and the fourth termon the right-hand sides of (3) and the third term on the righthand sides of (4) represent the rate of natural generation.
Figure 1 illustrates these mutual relationships.
Next, with regard to the diffusion terms of the reactiondiffusion equations (1), the activator-inhibitor model requires
the following relationshipκh > κa,(5)

meaning that the inhibitor spreads faster and more widely
than the activator.
In the activator-inhibitor model, when the reaction term
and the diffusion term interact appropriately, the following
situations occur.
• The activator present at a position generates more activator and the inhibitor at the same position.
• At that time, the inhibitor quickly diffuses around theposition and suppresses the increase of the activator in
the surroundings.
• When the activator is suppressed in the surroundings,
not only the activator but also the growth of the inhibitor

Turing patterns generated by the activator-inhibitor system.

is suppressed.
• When the inhibitor decreases, both the activator andthe inhibitor proliferate by the action of the activator,
and the generated inhibitor diffuses to its surroundings
more quickly than the activator.
By repeating this process, the spatial distributions of the activator and the inhibitor become uneven and appear as geometric patterns in the space. This is called the Turing pattern.
Figure 2 shows an example of the evaluation of the behaviorof the activator and the inhibitor by the activator-inhibitor
model in one-dimensional space [21]. We can recognizethat a periodic pattern appears in the spatial distributions ofboth the activator and the inhibitor with the passage of time.
3.

Activator-Inhibitor Model for Fake News and Corrections

3.1

Basic Concept

Based on the case of toilet paper hoarding shown in the Introduction, we propose a mathematical model to understandthe mechanism by which the spread of both fake news andthe corrections results in toilet paper hoarding. In preparation, let us consider the basic properties of fake news and thecorrections and their interaction. We assume the following
properties as a starting point for the model.
• The spread of the corrections fuels users’ anxiety that
toilet paper may be hoarded by many people contrary to
the original purpose of the corrections. For this reason,
the corrections not only convey their original intent but
also attract attention to the fake news.
• The spread of fake news makes the content of fake news
known, and it becomes difficult to spread the corrections
that deny it.
• In general, fake news deals with topics that are of interest to ordinary users and therefore tend to spread fasterover OSNs than regular news [17].
The first item in the bullet point list shows that the spreadof the correction has a role in inspiring attention to not only
the correction itself but also the fake news. The second item
shows the property that the spread of fake news suppresses

AIDA et al.: CLUSTER STRUCTURE OF ONLINE USERS GENERATED FROM INTERACTION BETWEEN FAKE NEWS AND CORRECTIONS

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where A is the adjacency matrix defined as 
A := wi j 1≤i, j ≤n ,and D is the weighted degree matrix defined as
D := diag(d1, . . . , dn ).
Here, di is the weighted nodal degree of node i defined asdi :=Fig. 3

Interaction between fake news and their corrections.

• activator: corrections• inhibitor: fake news
Figure 3 shows the structure of this relationship corresponding to that of Fig. 1.
Activator-Inhibitor Model on Networks

In this section, we propose an activator-inhibitor model that
describes the interaction between fake news and correctionsin OSNs, based on the concept shown in Sect. 3.1.
Consider undirected graph G(V, E) with n nodes representing OSN structure, where V = {1, 2, . . . , n} is the setof nodes and E is the set of undirected links. In terms ofcorrespondence with OSN, node i ∈ V represents a user, andundirected link (i, j) ∈ E represents the relationship between
user i and user j.
Originally, the strength of relationships between users
in OSNs is asymmetric, so it is natural to model the structure of OSNs with a directed graph. However, in this paper,
we would like to discuss the phenomenon that information
dispersal becomes biased even though the network structure
itself is not biased at all. For this reason, we dare to introduce a highly symmetric network model represented by anundirected graph. As a side note, it is easy to extend the
network model to directed graphs and to discuss information
spreading on them, as there are no technical problems indoing so. However, it is worth noting that directed graphs
yield asymmetric effects on the networks.
We assume that each link in G(V, E) has a link weightthat represents the strength of the relationship; let the weightof undirected link (i, j) ∈ E between node i and node j
be wi j > 0. For convenience, the weight of the node pair(i, j) < E is assumed to be wi j = 0.
The Laplacian matrix L representing the structure ofG(V, E) is defined as
L := D − A,wi j .

i=1

the activity of correction, and the last bullet point meansthat the diffusion coefficient of fake news is larger than the
diffusion coefficient of the correction. Based on the above
properties, the following correspondence can be assumed in
the activator-inhibitor model.

3.2

n
Õ

(6)

The Laplacian matrix L can be used to describe thediffusion phenomenon on the network. Let xi (t) (≥ 0) be thestate quantity of node i at time t, and the state vector x(t) be
defined asx(t) := t(x1 (t), . . . , xn (t)).
The diffusion equation on the network can, by using the
Laplacian matrix, be expressed asdx(t) = −L x(t).
dtWhen considering the activator-inhibitor model on the
network, the diffusion term appearing in the activatorinhibitor model can be expressed by using the Laplacianmatrix. The activator-inhibitor model on the network is expressed as follows. Let ai (t) and hi (t) be the amounts ofactivator and inhibitor at node i at time t, respectively. Next,
we define two types of state vectors,
a(t) := t(a1 (t), . . . , an (t)),h(t) := t(h1 (t), . . . , hn (t)),(7)

which have the amounts of activators and inhibitors for all
nodes as elements. Then, the equations that describe theactivator-inhibitor model on the network can be written as∂a(t) = F(a(t), h(t)) − κa L a(t),∂t∂h(t) = G(a(t), h(t)) − κh L h(t).
∂t(8)

Here, F(a(t), h(t)) and G(a(t), h(t)) are functions that express the reaction terms, and −κa L a(t) and −κh L h(t) are
the diffusion terms. By following (3) and (4), the functions
representing the reaction terms are introduced as
F(a(t), h(t)) := Ca a(t) − Sa h(t) − Ma a(t) + Ra 1,
G(a(t), h(t)) := Ch a(t) − Mh h(t) + Rh 1.
(9)
Here, Ca , Ch , Ma , Mh , Ra , and Rh are all diagonal matrices
defined as
Ca := ca I = diag(ca, . . . , ca ),Sa := sa I = diag(sa, . . . , sa ),Ma := µa I = diag(µa, . . . , µa ),IEICE TRANS. COMMUN., VOL.E106–B, NO.5 MAY 2023

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Mh := µh I = diag(µh , . . . , µh ),Ra := ρa I = diag(ρa, . . . , ρa ),Ch := ch I = diag(ch , . . . , ch ),Rh := ρh I = diag(ρh , . . . , ρh ),where I is an n × n unit matrix, and 1 is an n-dimensional
vector 1 := t(1, . . . , 1). The fact that these matrices are
diagonal matrices reflects the assumption that the interaction
between the activator and the inhibitor is a local interaction
at each node.
In the above model, for the sake of simplicity and higher
symmetry, all the above diagonal matrices are set to be proportional to the identity matrix. This corresponds to allnodes having the same parameter value. More generally, we
can use a diagonal matrix where each node has a different
parameter value.
4.

Experimental Evaluation for User Cluster Generation

This section performs numerical experiments of theactivator-inhibitor model that describes the interaction between fake news and the corrections on a homogeneous network that does not have a cluster structure. The purpose of
the experimental evaluations is to show that even if there are
no clusters in the network structure itself, clusters of usersthat are strongly influenced by fake news might be formedon OSNs. Therefore, we conduct experiments on networkmodels with extremely high symmetry.
The experimental environment is described as follows.
We use a two-dimensional lattice network model consistingof 10, 000 nodes in a 100×100 configuration, with a periodic
boundary condition that yields a torus topology (Fig. 4). All
links are weighted at 1. Other parameters are summarizedin Table 2. The parameter values were determined withreference to the Reaction-Diffusion System Simulator [22]published on the website of Professor Shigeru Kondo’s Laboratory, Osaka University. In addition, the values of ai (t)and hi (t) are restricted to the range of [0, 1]. Figures 5 and 6
show the initial conditions for two different conditions. The
dark part indicates where the activator value is large, and thelight part indicates where the activator value is small.
Figure 5 shows the situation that the correction of the
fake news starts to spread from the center nodes after the
fake news has spread widely in advance. The left panel isthe initial distribution of the correction which exists onlyat the center nodes; the right panel is the initial distribution of the fake news that has been uniformly distributed.
This situation mirrors the actual toilet paper case. The fakenews was widely distributed in advance but not particularly
well noticed; the correction was subsequently spread from
specific nodes. On the other hand, Figure 6 shows the situation that both the fake news and its correction have been
widely distributed by mass media, many users have seen andbeen influenced by both pieces of information, more or less.
Both the left and right panels are the initial distributions of
the correction and the fake news that have been randomly

100 × 100 2-dimensional lattice network model with torus bound-Fig. 4ary.

Table 2
ca
0.08

sa
0.08

Parameter setting for the activator-inhibitor model.
µa
0.03

ρa
0.0015

κa
0.02

ch
0.11

µh
0.12

ρh
0.0005κh0.5

Fig. 5 The initial condition of the distributions of the strengths of thecorrection (left) and the fake news (right): the correction starts to propagate
from the center nodes, and the fake news has been uniformly distributed inadvance.

Fig. 6 The initial condition of the distributions of the strengths of thecorrection (left) and the fake news (right): both are randomly distributed
between [0.0, 0.1].

distributed in the range of [0, 0.1].
Figure 7 shows the distributions of the correction andthe fake news for the initial condition of Fig. 5 after sufficient time has passed. The difference in the intensity of thecorrection (the value of activator) directly corresponds tothe difference in the intensity of the fake news (the value of
inhibitor). That is, where the intensity of the correction islarge, the intensity of the fake news is also large. Conversely,
where the intensity of the correction is small, the intensity ofAIDA et al.: CLUSTER STRUCTURE OF ONLINE USERS GENERATED FROM INTERACTION BETWEEN FAKE NEWS AND CORRECTIONS

397

from occurring. In order to find the solution, it is necessary tounderstand what kind of conditions cause the Turing pattern.
Here, based on the reaction term (9) of the activator-inhibitor
model, we discuss Turing instability, which is a necessary
condition for the Turing pattern to occur.
From reaction term (9), the ith components (1 ≤ i ≤ n)of F(a(t), h(t)) and G(a(t), h(t)) are expressed, respectively,
asFig. 7 Turing patterns generated by the interaction between fake news
and the correction: the distributions of the strengths of the correction (left)
and the fake news (right) for the initial condition of Fig. 5.

f (ai (t), hi (t)) = ca ai (t) − sa hi (t) − µa ai (t) + ρa,g(ai (t), hi (t)) = ch ai (t) − µh ai (t) + ρh .
(10)Fig. 8 Turing patterns generated by the interaction between fake news
and the correction: the distributions of the strengths of the correction (left)
and the fake news (right) for the initial condition of Fig. 6.

the fake news is also small. From the correspondence withfake news, we find that fake news is activated in areas where
the correction is actively discussed, while fake news does
not attract attention in places where the correction did notreceive attention. The important point of this experimental
result is that the correction did not uniformly reduce the influence of fake news, but triggered differentiation yielding
areas where fake news is strongly noticed and parts where it
is not.
Figure 8 shows the distributions of the correction andthe fake news for the initial condition of Fig. 6 after sufficient
time has passed. The Turing patterns appearing on the network are different from those of Fig. 7, but we have similar
results in that the correction did not uniformly reduce the influence of fake news, but triggered the generation of clusters
of users actively influenced by the fake news.
Since the network model used in the experiment is a
network model with very high symmetry (a homogeneous
undirected graph with lattice topology) and periodic boundary conditions, the network structure has no location-specificproperties. Therefore, the polarization that appears in theexperimental results is the heterogeneity created by the interaction of fake news and the correction. In this way, theactivator-inhibitor model can qualitatively explain the phenomenon in which the corrections cause a situation in whichfake news receives more attention from users.
5.

Condition for Preventing User Cluster Generation

In order to prevent the generation of user clusters activated by
the interaction of fake news and corrections, it is necessary to
prevent the Turing pattern corresponding to the user clusters

The necessary conditions for the Turing pattern to occur
include two factors: uniform solutions are stable if there are
no diffusion effects, otherwise they become unstable.
First, when there is no diffusion (assuming the diffusion
coefficients are κa = κh = 0), the spatially-uniform distribution needs to be a stable steady solution of (1). Since there
is no diffusion, it is not necessary to consider spatial uniformity, so it is sufficient to derive the conditions for stabilizing
the steady-state solution. In (8), by setting κa = κh = 0,
relations of the ith components are extracted as∂ai (t) = f (ai (t), hi (t)),∂t∂hi (t) = g(ai (t), hi (t)).
∂t

(11)

If small deviations from the steady-state solution ā and h̄ are∆a and ∆h, respectively, f and g are expanded asf (ā + ∆a, h̄ + ∆h)∂ f (ā, h̄)∂ f (ā, h̄)∆a +∆h + · · · ,∂a∂hg(ā + ∆a, h̄ + ∆h)= f (ā, h̄) += g(ā, h̄) +∂g(ā, h̄)∂g(ā, h̄)∆a +∆h + · · · .
∂a∂hSubstituting the above into (11), and applying the steadystate solution condition∂ ā= f (ā, h̄) = 0,∂t
∂ h̄
= g(ā, h̄) = 0,∂t
we obtain∂(ā + ∆a) ∂(∆a)=∂t∂t∂ f (ā, h̄)∂ f (ā, h̄)=∆a +∆h + · · · ,∂a∂h∂( h̄ + ∆h) ∂(∆h)=∂t∂t∂g(ā, h̄)∂g(ā, h̄)=∆a +∆h + · · · .
∂a∂hBy linearizing the above equations, we obtain

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   ∂ f (ā, h̄)∂ ∆a=  ∂a ∂g(ā, h̄)
∂t ∆h

 ∂a

∂ f (ā, h̄)   ∂h  ∆a .
∂g(ā, h̄)  ∆h

∂h 

(12)

The condition that the steady-state solution t(ā, h̄) is stable
is that all eigenvalues of the matrix on the right side of (12)are negative. The following two equations can be obtained
as conditions for this.
∂ f (ā, h̄) ∂g(ā, h̄)
+
< 0,
∂a
∂h
∂ f (ā, h̄) ∂g(ā, h̄) ∂ f (ā, h̄) ∂g(ā, h̄)
−
> 0.
∂a
∂h
∂h
∂a

∂ f (ã, h̃)∂g(ã, h̃)
κh
+ κa
> 0,
(18)
∂a
∂h

2
∂ f (ã, h̃)∂g(ã, h̃)
κh
+ κa
∂a
∂h


∂ f (ã, h̃) ∂g(ã, h̃) ∂ f (ã, h̃) ∂g(ã, h̃)> 0.
− κa κh
−
∂a
∂h
∂h
∂a
(19)

(13)
(14)

Next, we consider the conditions for destabilizing thesteady-state solution due to the presence of diffusion. Whenthe effect of the diffusion term (general diffusion coefficientκa , κh > 0) is added, the spatial uniformity of the solution
must be broken and the uniform distribution is no longer astationary solution of (1). First, since the Laplacian matrix
L that appears on the right side of (8) is a symmetric matrix, it can, by using an appropriate orthogonal matrix P, bediagonalized as
P L P = Λ := diag(λ1, . . . , λn ).

tHere, due to the nature of the Laplacian matrix, it isλi ≥ 0 for all i (1 ≤ i ≤ n). Diagonalizing (8) using thisyields∂ t
P a(t) = F(tP a(t), tP h(t)) − κa Λ tP a(t) ,
∂t


∂ t
P h(t) = G(tP a(t), tP h(t)) − κh Λ tP a(t) .
∂t

The conditions (13)–(14) and (18)–(19) are known as conditions for Turing instability.
Next, we consider the concrete conditions that are derived from Turing instability conditions (13)–(14) and (18)–(19). From (3) and (4), we obtain∂f= ca − µ a ,∂a∂f= −sa,∂h∂g= ch ,∂a∂g= −µh .
∂hWe first consider the conditions for parameters other thanthe diffusion coefficients κa and κh . From (13) and (18), we
ontain0 < ca − µa < µh .

(20)

Also, from (14), we obtain(ca − µa ) µh < sa ch .
(15)


By introducing
the components of the vectors tP a(t) andtP h(t) respectively ast
P a(t) := t(ã1 (t), . . . , ãn (t)),
t
P h(t) := t( h̃1 (t), . . . , h̃n (t)),
we can extract the relationships of the ith components as∂ãi (t) = f (ãi (t), h̃i (t)) − κa λi ãi (t),∂t∂h̃i (t) = g(ãi (t), h̃i (t)) − κh λi h̃i (t).
∂t

In order for diffusion to destabilize a spatially-uniform
steady-state solution, the matrix appearing on the right-handside of (17) must have at least one positive eigenvalue. In therange of λi ≤ 0, as the sufficient conditions for the existenceof at least one positive eigenvalue in the above matrix, we
obtain(16)

Following the same procedure as (12), we consider small
variations ∆ãi and ∆ h̃i around ãi (t) and h̃i (t) corresponding
to spatially-uniform solutions. By linearizing it, we obtain∂ ∆ãi∂t ∆ h̃i ∂ f (ãi , h̃i )∂ f (ãi , h̃i )− κa λi ∆ãi∂a∂h=  ∆ h̃i .
∂g(ãi , h̃i )∂g(ãi , h̃i )− κh λi 



∂a
∂h
(17)

(21)

Next, we consider the conditions related to diffusion
coefficients κa and κh . After satisfying the conditions (20)(21), in order to satisfy (19) and (5), the following condition
is required:"1
κa < κh 2 − (ca − µa ) µh + 2 sa ch
µh
#
q
2
2
− 2 sa ch − sa ch (ca − µa ) µh .
Therefore, in order to block the generation of the Turing
pattern, the diffusion coefficient κa of the activator should
be set as"1
κa ≥ κh 2 − (ca − µa ) µh + 2 sa ch
µh
#
q
2
2
− 2 sa ch − sa ch (ca − µa ) µh .
(22)6.

Experimental Evaluation of the Condition for Preventing User Cluster Generation

The lower limit of the diffusion coefficient κa of the activator

AIDA et al.: CLUSTER STRUCTURE OF ONLINE USERS GENERATED FROM INTERACTION BETWEEN FAKE NEWS AND CORRECTIONS

399

Fig. 9 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 5 with κ a = 0.057.

Fig. 12 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 6 with κ a = 0.057.

Fig. 10 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 5 with κ a = 0.058.

Fig. 13 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 6 with κ a = 0.058.

Fig. 11 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 5 with κ a = 0.059.

Fig. 14 The distributions of the strengths of the correction (left) and thefake news (right) for the initial condition of Fig. 6 with κ a = 0.059.

given by (22) gives sufficient conditions to prevent the Turing
pattern from occurring. From the values in Table 2, the lower
limit of κa in the experiments of this paper is determined asfollows:
κa ' 0.058.

(23)

In this section, we investigate the occurrence of Turing patterns around this value and consider the effectiveness ofcondition (22) in countering fake news.
Figures 9–11 show the distributions of correction (leftpanels) and fake news (right panels) for the diffusion coefficient κa = 0.057, 0.058, and 0.059 after sufficient time has
passed, for the initial condition illustrated in Fig. 5. This initial condition describes the situation in which the correction
is sent from the central node after the fake news has spreadwidely. The result of Fig. 9 is when the diffusion coefficientκa = 0.057 does not satisfy condition (22), and the partial
emergence of the Turing pattern can be recognized. On theother hand, the results of Figs. 10 and 11 correspond to thecase where condition (22) is satisfied, and the Turing pattern
has not emerged.
Next, we show similar evaluations for a different initialcondition. Figures 12–14 show the distributions of correction (left panels) and fake news (right panels) for the diffusion
coefficient κa = 0.057, 0.058, and 0.059 after sufficient time
has passed, for the initial condition illustrated in Fig. 6. The
result of Fig. 12 is when diffusion coefficient κa = 0.057
does not satisfy condition (22), and the slight emergence of
the Turing pattern can be recognized. On the other hand,
the results of Figs. 13 and 14 correspond to the case wherecondition (22) is satisfied, and the emergence of the Turing
pattern is blocked.
From these results, it can be seen that condition (22),which blocks the Turing pattern, gives a highly accurate
threshold value for the presence or absence of the Turing
pattern; it is a sufficient condition. It can also be seen thatits effectiveness does not depend on the initial conditions.
The value of κa is related to the speed at which thecorrection spreads. It might be generally difficult to know
the values of the parameters κh , ca , µa , sa , ch , and µh thatappear on the right-hand side of condition (22). Therefore,
it might be difficult to determine the value on the right sideof condition (22), and it may not be possible to know thelower limit of κa . However, there is no doubt that the occur-

IEICE TRANS. COMMUN., VOL.E106–B, NO.5 MAY 2023

400

rence of the Turing pattern can be suppressed as long as κais increased until it exceeds a certain value. For this reason,
there is no doubt that these results set the direction of countermeasures against fake news. Moreover, the conditions do
not depend on the initial state of the network, and in theory,
they do not depend on network topology, so the versatility
and utility of these results are expected to be high.
The structures of existing online social networks are, in
general, asymmetrical and might be another factor generating user clusters. Asymmetrical structures may be due to, inpart, directed links, differences in link weights, and the original clustered structure of the social network. Therefore, if
we want to know a general method to prevent the generationof user clusters, we should investigate the applicability of thecondition for avoiding user cluster generation to the generalasymmetric networks; however, this issue is out of the scopeof the paper.
7.

Conclusion

In this paper, we proposed a model that explains the occurrence of the phenomenon in which the information issued
to correct fake news on OSNs does not always yield the expected effect, and may, in fact, worsen the influence of fakenews. By using the activator-inhibitor model, which is akind of reaction-diffusion system, and making the inhibitor
correspond to fake news and the activator correspond to thecorrections, we were able to well replicate the characteristics and interactions of both. Numerical experiments showedthat even if there is no bias in the network structure itself,
the interaction between fake news and the corrections may
trigger the generation of user clusters that are strongly influenced by fake news. In addition, by using linear stability
analysis, we derived the sufficient condition for the diffusion
coefficient of the correction that prevents the generation ofuser clusters affected by fake news. Although it is a sufficientcondition, experimental results show it gives a sufficiently
accurate threshold to the generation of user clusters. In addition, the condition is independent of the initial conditionof the distributions for the fake news and its correction. If
the network structure has lower symmetry, user clusters canbe generated for other reasons related to the asymmetry of
the structure. Asymmetric structures can be generated bydirected links, differences in the link weights, originally biased structures of the social network, and so on. Therefore,
preventing user clusters requires the consideration of various
factors. However, even if we eliminate the issues raised byasymmetric structures, user clusters are still generated by
the mechanism discussed in this paper. In the future, basedon the proposed activator-inhibitor model, we will consider
effective and strategic methods for transmitting corrections
to counter fake news that are applicable to general network
structures.
Acknowledgments
The authors would like to thank Mr. Satoshi Furutani at

NTT Labs. for useful discussions. 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), and Grant-in-Aid for Challenging Research (Exploratory) No. 21K19775 (2021–2023)
from the Japan Society for the Promotion of Science (JSPS),
and TMU local 5G research support.
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html

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.

Takumi Sakiyama
received his B.E. degreein Computer Science from Tokyo Metropolitan
University, Japan, in 2022. Currently, he is a
graduate student of the Graduate School of Systems Design, Tokyo Metropolitan University.

Ayako Hashizume
received Ph.D. in Kansei Science from Tsukuba University, Japan, in
2011. She is an associate professor at Hosei University, Japan. From 2012 to 2018, she was an
assistant professor at Tokyo Metropolitan University, Japan. Her research interests include
psychology of emotion and Kansei engineering
in relation to HCD (human centered design) andUX (user experience), especially user surveys
for service design. She was awarded the kids
design prospective competition 2008 from theJapan Kids Design Association, the research prize 2013 from the Funai
Foundation for Information Technology, and the best book award 2020 and
2021 from Japan Society of Kansei Engineering. She is also a member of
ISO/TC 159/SC 4.

Chisa Takano
received a B.E. degree
in Telecommunication Engineering from Osaka
University, Japan, in 2000, and a Ph.D. in System Design Engineering from Tokyo Metropolitan University, Japan, in 2008. In 2000, shejoined the Traffic Research Center, NTT Advanced Technology Corporation. From April
2008 to March 2020, she was an Associate Professor at the Graduate School of Information Sciences, Hiroshima City University. She has beena Professor of the Graduate School of Information Sciences, Hiroshima City University since April 2020. Her researchinterests are computer networks, distributed systems and social networks.
She received the IEICE Young Researchers’ Award in 2003. She received
the Information Network Research Awards from the IEICE in 2004, 2012,2015 and 2019. She is a member of IEEE, IEICE and IPSJ.