NOLTA, IEICE
Paper

Evaluation of user dynamics created byweak ties among divided communities
Takahiro Kubo 1 a) , Chisa Takano 2 , and Masaki Aida 1
1

Graduate School of Systems Design, Tokyo Metropolitan University,
Hino, 191-0065 Japan

2

Graduate School of Information Sciences, Hiroshima City University,
Hiroshima, 731-3194, Japana)kubo-takahiro@ed.tmu.ac.jp

Received October 10, 2020; Revised December 9, 2020; Published April 1, 2021
Abstract: Flaming phenomena represent the divergence in the strength of user dynamics as
created by user interactions in online social networks (OSNs). Although it has been known
that ﬂaming phenomena occur when the Laplacian matrix of the OSN has non-real eigenvalues,
it was recently shown that ﬂaming phenomena may occur even if all the eigenvalues are real
numbers. This eﬀect appears only in the situation that some eigenvalues are degenerate, and a
special unitary transformation is applied to the equations representing user dynamics; whether
actual OSNs satisfy this condition has not been fully discussed. In this paper, we clarify thatthe user dynamics caused by the degeneration of eigenvalue 0 is one speciﬁc example of theabove condition. We also investigate the mechanism and characteristics of ﬂaming phenomena generated by degenerated eigenvalues. Furthermore, we demonstrate through numericalsimulations that the degeneration of eigenvalues can cause divergence.
Key Words: online social network, wave equation, ﬂaming, phase synchronization

1. Introduction
The development and spread of information networks have dramatically activated interpersonal communication and the ability of individuals to disseminate information; they have revolutionized the way
we do business and live our daily lives. In particular, various social networking services (SNSs) provided on the Internet support the activation of social networks. We call the relationship between suchusers an online social network (OSN). Activation of OSNs is the foundation for highly eﬃcient social
activities, and for this reason, SNSs are indispensable in modern society. However, we cannot ignore
the fact that OSNs also have the risk of causing social instability. The existence of SNSs has playeda highly negative role in international terrorist activities and the intensiﬁcation of social divisions
based on ideological, political, and religious disagreements. While information network technologies
contribute to the eﬃciency of social activities, they can also accelerate the destabilization of social
systems.
Typical of the negative impacts on real-world activities possible is the ﬂaming phenomena. In [1, 2],
the deﬁnition of ﬂaming phenomena is drawn from observations of actual network services. For exam-

157
Nonlinear Theory and Its Applications, IEICE, vol. 12, no. 2, pp. 157–174c
IEICE
2021

DOI: 10.1587/nolta.12.157

ple, [2] deﬁnes ﬂaming phenomena as ‘displaying hostility by insulting, swearing or using otherwise
oﬀensive language’ from observations of comments generated in an actual YouTube service. In suchsituations, it leads to the activation of user discussions and/or the escalation of user conﬂicts. Ourdeﬁnition of ﬂaming phenomena, based on an analytical model of user dynamics, does not contradict
the observation-based deﬁnition. The activation or escalation can be represented as phenomena that
increases the intensity of user dynamics. Therefore, in our framework, the deﬁnition of the ﬂaming phenomena is that the strength of user dynamics on OSNs diverges as users interact with eachother. It is diﬃcult to provide an immediate and fundamental solution to the ﬂaming phenomena
because their causes are rooted in various social characteristics and problems. However, if we leave
the phenomena until a full solution is found, there is a risk that the spread of ﬂaming phenomena will
yield severely negative social eﬀects. Therefore, it is desirable to develop mitigation techniques for
ﬂaming phenomena that can be implemented immediately, without addressing the underlying social
causes. It is desirable to understand the system’s characteristics common to ﬂaming phenomena andto consider countermeasures, rather than trying to analyze the speciﬁc factors that cause each caseof ﬂaming. A model proposed to understand the engineering mechanism of ﬂaming phenomena thatinvolves OSN structure is the oscillation model [3]. This model describes user dynamics in terms of
the wave equations deﬁned on the OSN, and can not only describe user dynamics as a solution to thewave equation but also explicitly describe the causal relationship between the structure of the OSN
and user dynamics.
It is known that the network oscillation model predicts that ﬂaming phenomena occur when the
Laplacian matrix, which represents the structure of OSN, has non-real eigenvalues. This enables us
to understand the causes of ﬂaming phenomena in relation to the network structure and provides aframework for considering countermeasures to ﬂaming phenomena within the framework of network
control technology. The conventional understanding is that ﬂaming phenomena could not occur when
all eigenvalues of the Laplacian matrix representing the OSN structure were real numbers.
Recently, however, [4] showed that even if all eigenvalues of the Laplacian matrix representing
the OSN structure were real numbers, ﬂaming phenomena can still occur under the following two
conditions.
• When the Laplacian matrix cannot be diagonalized.
The prerequisite for the inability to diagonalize the Laplacian matrix is that some of the eigenvalues of the Laplacian matrix must be degenerate. However, diﬀerent from a purely mathematicalmodel, it is extremely unlikely that the eigenvalues would be degenerate in a Laplacian matrix
of a complex network structure, such as the structure of an OSN. This corresponds to the factthat the eigenvalue equation has multiple roots. Considering that, in general, it is extremely
unlikely that a n-th order equation with randomly determined coeﬃcients will have multiple
solutions, we can understand that this is a condition in which duplication of eigenvalues is extremely unlikely. However, there is an exception wherein duplication of eigenvalues can readily
occur, which we discuss later.
• An equation with a speciﬁc unitary transformation is used instead of the original equation todescribe user dynamics.
In the oscillation model, the user dynamics are described by wave equations deﬁned on the
network. This (original) wave equation is invariant to the unitary transformation. However, if
we extend the wave equation to explicitly describe the causal relationship between the network
structure and user dynamics, the extended equations are not invariant to unitary transformations. This means that we have diﬀerent extended equations depending on whether the starting
point is the original wave equation or the unitary-transformed wave equation. The user dynamics derived using the unitary-transformed equation may diverge even when all eigenvalues of
the Laplacian matrix are real numbers. However, the unitary-transformed equation is diﬀerent
from the original wave equation and therefore is generally diﬀerent from the phenomena that
occur in the real world. However, there is an exception with regard to the extended equations
being invariant for the unitary transformation. We will discuss its user dynamics later.

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In social network analysis, there are many examples of relationships that are not usually associatedwith each other, but sometimes provide important “insights” that lead to advances in the future.
For example, a person with whom one does not normally have much contact may bring unexpected
information about a new job, or an exchange between researchers in diﬀerent ﬁelds may provide asolution to an intractable problem. Such a link, which is not strong, but does play an importantrole, is called a “weak tie”. From this point of view, the phenomenon that occurs when normallyseparated communities are linked for some reason is an interesting subject of study with regard to
the user dynamics of social networks. It is known that the Laplacian matrix representing the network
structure always has an eigenvalue of 0, and the degenerate number of eigenvalue 0 is equal to thenumber of connected components of the network. Hence, for an entire connected network, there is only
one eigenvalue 0 in the Laplacian matrix representing its structure, but for a network split into twoparts, there are two duplicate eigenvalue 0 instances in the Laplacian matrix representing its structure.
As mentioned in the ﬁrst bullet point above, the degeneration of eigenvalues of the Laplacian matrix
is a very unlikely situation for eigenvalues other than 0. However, it is not uncommon for overlap to
occur in terms of eigenvalue 0; it represents a split into multiple isolated networks. This corresponds
to the emergence of a fragmented community in OSNs. Furthermore, in relation to the second bullet
point above, there is only one case where the equation explicitly describes the causal relationshipwith the original equation, even with the unitary transformation, and that is when we consider a partof the equation corresponding to zero eigenvalues. Thus, user dynamics in a situation where somechanges occur in divided communities that are linked by “weak ties” can be considered to correspondto the typical case for the situation assumed in the literature [4].
In this paper, we consider user dynamics generated from weak interactions between divided communities as one speciﬁc example of the mechanism of ﬂaming phenomena shown in [4]. Speciﬁcally, we
study the characteristics of the user dynamics generated by the interaction of the oscillation modes,
which are degenerate with eigenvalue 0 and interact with each other in Jordan canonical form. This
paper is organized as follows. Section 2 presents related research on the user dynamics of OSNs andclariﬁes the position and importance of this research. Section 3 describes the modeling of user dynamics arising from the weak interaction between divided communities and examines the propertiesof user dynamics that arise from them. In Section 4, we show the validity of the ﬂaming phenomena
derived in Section 3 by means of numerical simulations. Finally, we conclude this paper in Sec. 5.

2. Related work
Since it is easier to obtain structural data for online social networks than real-world social networks,
research on network analysis has been very active [5–7]. Online social networks can be roughlyclassiﬁed into two types: social relationship networks [8–12], which are formed by the relationshipbetween followers, and social interaction networks [13–16], which are the actual interactions between
peers. To deeply understand the structure of online social networks, it is necessary to apply theoreticalmodels in addition to analyzing real data.
Studies on theoretical models of online social networks can be summarized from the following pointsof view.
There are two main models for explaining the diﬀusion of a new service or the spread of a rumor
on an OSN. One is macroscopic modeling, which does not directly describe the state of individual
users, while the other is microscopic modeling, which deals directly with the state of each individualuser. The macroscopic model focuses on the size of the set of users with speciﬁc characteristics, suchas the number of service subscribers or the number of people who know about the rumor, while themicroscopic model focuses on the state of individual users [17, 18].
With regard to the macroscopic model, studies have detailed the SIS and SIR models of infectious
diseases, in which service subscribers and people who are aware of rumors are likened to those whoare infected with an infectious disease [19–22]. These models describe the dynamics of infection byrepresenting the time variation of macroscopic quantities in the network as ﬁrst-order diﬀerential
equations. The SIS and SIR models do not model the dynamics of individual users and do not need
to be aware of the microscopic state of individual users. In these models, explosive user dynamics

159

refers to a high rate of state change (e.g., rapid propagation of rumors) and does not deal with thedivergence of the strength of user dynamics, e.g. ﬂaming phenomena.
As microscopic models, studies using percolation models have described individual user dynamics [23–26]. These models show how the state change spills over to the whole system through the
dynamics of individual user state transitions. As well as percolation, the Markov chain model isknown as a model that can deal with the dynamics of individual user state [17, 18, 27–30]. Song etal. modeled the diﬀusion of innovation in the network (NW) using continuous-time Markov chainsto adjust the speed of ﬂow between individuals to account for communication delay [17]. On theother hand, Leenders et al. expanded the possibilities for longitudinal binary social network data to
be treated as continuous-time Markov chains [18]. Modeling based on continuous-time Markov chainscan describe individual user state changes, but it cannot describe the divergence of user dynamics,
such as ﬂaming phenomena, because it considers only the steady-state condition and the transientstate leading to it.
In this study, we deal with wave equations on networks [3, 31, 32]. This reﬂects the fact that theinﬂuence between OSN users should propagate at a ﬁnite velocity. The model is capable of representing
the state of individual users and describing the divergence in the strength of user dynamics, such as
ﬂaming phenomena.

3. Degenerated oscillation model in multiple isolated networks
3.1 User dynamics when the eigenvalues degenerate
In this subsection, we introduce the oscillation model on the network that describes how users interact
with each other on an OSN.
The oscillation model on networks is a minimal model for describing the user’s state. It also rulesof interaction between users, as simply and as universal as possible. We describe the state of a user
by a one-dimensional variable, xi (t), which is the state of user i at time t, where i = 1, 2, . . . , n and
n is the number of users. The deﬁnition of the state vector for all users is written asx(t) := t(x1 (t), . . . , xn (t)),(1)

where superscript t represents the transposition of the state vector. Let us consider the interaction
between users. User i receives a force, Fij , from neighboring user j that makes |xi (t) − xj (t)| smaller.
The strength of the force is proportional to xi (t) − xj (t) and is written as follows
Fij = −wij (xi (t) − xj (t)),(2)

where wij ≥ 0 is a constant. Since the interaction between users is generally asymmetric, wij = wji .
Then, the equation of motion of the state vector of users is written as the following wave equation:d2x(t) = −L x(t),
dt2

(3)

where L is the Laplacian matrix of the directed graph whose weight of directed link from i to j isgiven by wij . Although several deﬁnitions of the Laplacian matrix are known, we follow the deﬁnitionin [3]. Note that we assume all the eigenvalues of L are real numbers because we concentrate thediscussion on the generation of ﬂaming phenomena under this condition.
First, let us consider the case where Laplacian matrix L is diagonalizable. By applying a transformation such that L is diagonalized, we transform the wave equation (3) intod2
φ(t) = −Λ φ(t),
dt2

(4)

where Λ is the diagonalized Laplacian matrix, which is Λ := P −1 L P given the appropriate squarematrix P . In addition, φ(t) := P −1 x(t). The wave equation (3) on OSNs can be decomposed into n
independent equations for oscillation modes. Then, the solutions of (3) are given as follows:φ(t) = exp[∓i Ω t] φ(0),

160

(5)

where, for the eigenvalues of Laplacian matrix λμ (μ = 0, 1, . . . , n − 1), Ω can be deﬁned asΩ := diag
λ0 , λ1 , . . . , λn−1 .
If L is not diagonalizable, then the wave equation (3) is transformed into the form of (4) by using
the Jordan canonical form of L, Λ. The transformed wave equation (4) using Jordan canonicalform cannot be decomposed into independent oscillation modes due to the eﬀect of the non-diagonalcomponent of Λ. Therefore, in order to clarify the inﬂuence of the non-diagonal component, we
introduce the fundamental equation of networks [3, 32].
First, let us introduce an n × n matrix H that satisﬁes
H 2 = L.

(6)

Then, we can write the fundamental equation of (3) asd x(t)= H x(t).
dt

±i

(7)

The solution of (7) also satisﬁes the original wave equation (3). If L is not diagonalizable, then H is
not diagonalizable either. For this reason, we transform the fundamental equation (7) so that H isrepresented in Jordan canonical form Ω as follows±id φ(t)= Ω φ(t).
dt(8)

In (8), Ω is expressed as follows
⎡
⎢
⎢
⎢
⎢
⎢
⎢
Ω=⎢
⎢
⎢
⎢
⎢
⎢
⎣

..

⎤

.
ωμ

1
ωμ

0

1
..
.

⎥
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎥
⎦

0
1
ωμ
..

(9)

.

where diagonal component ωμ represents an oscillation mode. Let us focus on the example of the
Jordan block of oscillation mode ωμ , shown below, as it cannot be divided further.
⎡
⎤
ωμ 1
0
⎢
⎥
(10)
ΩJ = ⎣ 0 ωμ 1 ⎦ .
00 ωμThe Jordan block is an m × m matrix when the multiplicity of the degenerate eigenvalues is m. In(10), the ΩJ has m = 3. Note that the result of the matrix with m = 3 can be extended to thematrix with m > 3. Because in the Jordan block, even if the matrix size increases, the coupling rules
between adjacent oscillation modes are the same. Then, the fundamental equation is expressed using(10) as follows±id ψ(t)= ΩJ ψ(t).
dt

(11)

where ψ(t) is the solution of the fundamental equation (11).
Next, we decompose ΩJ into a constant multiple of the unit matrix Ω0J , and a matrix of othercomponents ΩIJ , as
ΩJ = Ω0J + ΩIJ .

161

(12)

We introduce Ω0J and ΩIJ to separate ΩJ into diagonalizable matrix with well known network properties
and others. Therefore, Ω0J and ΩIJ are determined by above the choosing rule. The fundamental
equation with the network structure of linear operator (12) can be expressed as follows±id ψ(t)= Ω0J + ΩIJ ψ(t).
dt

(13)

We then consider Ψ0 (t) and ψI (t) that satisfy the following equationsd Ψ0 (t)= Ω0J Ψ0 (t),dtd ψI (t)= Ψ0 (−t) ΩIJ Ψ0 (t) ψI (t),
±i
dt

±i

(14)
(15)

where Ψ0 (t) is a matrix with the initial conditions of Ψ0 (0) = I (the m × m unit matrix) and can be
expressed as follows
Ψ0 (t) = exp[∓i Ω0J t].

(16)

In the end, ψ(t) can be written as a solution in product form as follows
ψ(t) = Ψ0 (t) ψI (t).

(17)

Note that Ψ0 (t) is proportional to the unit matrix because the corresponding eigenvalues are degenerate. Therefore, Ψ0 (t) can commute with any m × m matrix. From the commutation relation thatΩJ Ψ0 (t) = Ψ0 (t) ΩJ , we obtain the following relation.
ΩJ = Ψ0 (t) ΩJ Ψ0 (−t).

(18)

We can write the fundamental equation (13) as follows±id ψ(t)= ΩJ ψ(t)dt= Ω0J + Ψ0 (t) ΩIJ Ψ0 (−t) ψ(t).

(19)

This means that the fundamental equation (13) is invariant under the unitary transform of ΩJ as
follows
ΩJ → Ψ0 (t) ΩJ Ψ0 (−t), ψ(t) → Ψ0 (t) ψ(t).
As a result, the structure of the oscillation mode is invariant with regard to the unitary transformgiven constant multiples of the unitary matrix, such as Ψ0 (t) and Ψ0 (−t).
Since L is a positive semideﬁnite matrix, the matrix H satisfying H 2 = L is unique if and only if it
is a positive semideﬁnite matrix. Similarly, Ω is unique for Λ if it is limited to a positive semideﬁnite
matrix. If we choose a matrix other than the positive semideﬁnite matrix, Ω is not unique as Ω2 = Λ.
Since the eigenvalues of L we deal with are real numbers, the real parts of the eigenvalues are nonnegative. This is guaranteed by Gerschgorin’s theorem [33]. Now, by choosing Ω appropriately, we
try to cast both Λ and Ω in Jordan canonical form. In the case of the Jordan block of interest,
consider that the square of ΩJ creates a structure of Jordan canonical form. In order to consider thenecessary conditions for this, we decompose the non-diagonalizable matrix ΩIJ of ΩJ into the diagonal
I(d)
I(a)
matrix ΩJ and the remainder matrix ΩJ as follows:I (d)ΩJ = Ω0J + ΩIJ = Ω0J + ΩJ

I (a)

+ ΩJ

The square of ΩJ is written as 2 
2 
2
0
I (d)
I (a)
Ω2J = ΩJ + ΩJ
+ ΩJ
I (d)

+ Ω0J ΩJ

I (d)

+ ΩJ

162

Ω0J

.

(20)

I (a)+ Ω0J ΩJ
I (d)

+ ΩJ

I (a)

+ ΩJ

I (a)

ΩJ

Ω0J

I (a)

I (d)

+ ΩJ

ΩJ

I (d)

Ω0J ,

I (a)

Ω0J ,

I (a)

ΩJ

.

(21)

If the following anti-commutation relations
I (d)

= −ΩJ

Ω0J ΩJ

I (a)

= −ΩJ

I (d)

I (a)

= −ΩJ

Ω0J ΩJ
ΩJ

ΩJ

I (d)

,

are established, all the cross-terms in (21) can be canceled, yielding correspondence in Jordan canonicalform as follows: 2 
2 
2
0
I (d)
I (a)
+ ΩJ
.
(22)
Ω2J = ΩJ + ΩJ
In order to achieve this, according to [4], we introduce the following Pauli matrices:



0 1
0 −i
1 0
σ1 =
, σ2 =, σ3 =
.
1 0
i 0
0 −1
For i, j = 1, 2, 3, the Pauli matrices satisfy the following relationships:
σi2 = E,σi σj = −σj σi ,(i = j).

where E is a 2 × 2 identity matrix. By using the Pauli matrices, we can introduce the following
matrices:Ω̂0J := Ω0J ⊗ σ3 ,
I (d)

Ω̂J

I (d)

:= ΩJ

⊗ σ1 ,

(23)

I (a)
I (a)
Ω̂J := ΩJ ⊗ σ2 ,where ⊗ represents the Kronecker product. By extending the matrix using the Pauli matrices, thesquare of Ω̂J can be expressed as 2 
2 
2
I (d)
I (a)
Ω̂2J = Ω̂0J + Ω̂J
+ Ω̂J
.
(24)
By using the Pauli matrices, we can rewrite the fundamental equation (13) as follows:d ψ̂(t)
I (d)
I (a)
±i
= Ω̂J ψ̂(t) = Ω̂0J + Ω̂J + Ω̂Jψ̂(t),
dt(25)

where Ω̂J is a 2m × 2m square matrix. ψ̂(t) is a 2m-dimensional vector deﬁned as↑↓ψ̂(t) = t(ψ1↑ (t), ψ1↓ (t), ψ2↑ (t), ψ2↓ (t), . . . , ψm(t), ψm
(t)).

(26)

Note that the extension of states ψ ↑ (t) and ψ ↓ (t) is an algebraic technique to simplify the computation.
Next, we introduce the unitary-transformed fundamental equation,



0
0
d ψ̂(t)  0
I (d)
I (a)
±i
= Ω̂J + e±i Ω̂J t Ω̂J + Ω̂Je∓i Ω̂J t ψ̂(t).
(27)
dt
as shown in [4]. The fundamental equation (11) for m dimensional state vectors is invariant with respect to the unitary transform. However, the fundamental equation (25) for the 2m-dimensional statevector is not invariant under unitary transformation since the fundamental equation (27) is diﬀerent
from (25). Since the fundamental equation describing the actual user dynamics is the equation (25)
before unitary transformation, the solution of Eq. (27) after the unitary transform does not necessarily
represent the actual user dynamics. However, since the unitary-transformed fundamental equation
has an interesting structure, our previous study [4] investigated the nature of the user dynamics described by the unitary-transformed equation and has revealed interesting properties that appear in
it.

163

3.2 Flaming phenomena caused by degenerate eigenvalues
One of the facts revealed in [4] is that the unitary-transformed fundamental equation of networks(27) describes divergence corresponding to the ﬂaming phenomena in spite of the fact that all the
eigenvalues are real numbers. This subsection brieﬂy explains its characteristics.
Figure 1 shows the state of three oscillation modes coupled as an example of (27). If the degenerate
I (d)
I (a)
eigenvalue in (27) is ω, then Ω̂0J , Ω̂J , Ω̂J can be expressed as follows
⎡
⎤
ω 0 0
⎢
⎥
Ω̂0J = Ω0J ⊗ σ3 = ⎣ 0 ω 0 ⎦ ⊗ σ3 ,
0 0 ω
⎡
⎤
d 0 0
⎢
⎥
I (d)
I (d)
Ω̂J = ΩJ ⊗ σ1 = ⎣0 d 0⎦ ⊗ σ1 ,
(28)
0 0 d
⎡
⎤
0 1 0
⎢
⎥
I (a)
I (a)
Ω̂J = ΩJ ⊗ σ2 = ⎣0 0 1⎦ ⊗ σ2 ,0 0 0where ω + d is the eigenfrequency of the degenerate oscillation mode.

Fig. 1.

Coupling states of 3 oscillation modes.

From (28), the linear operator in the fundamental equation (27) can be written as follows


0
0
I (d)
I (a)
e−i Ω̂J t
Ω̂0J + e+i Ω̂J t Ω̂J + Ω̂J


0
0
I (d)
I (a)
= Ω0J ⊗ σ3 + e+i ΩJ ⊗ σ3 t ΩJ ⊗ σ1 + ΩJ ⊗ σ2 e−i ΩJ ⊗ σ3 t⎡ω⎢ d e+i2 ω t⎢⎢= ⎢⎢⎢ 00⎢⎣
00d e−i2 ω t
−ω
0
0
0
0

0
ie

+i2 ω tω

d e+i2 ω t
00−i e−i2 ω t
0d e−i2 ω t
−ω
0
0

0
0
0
i e+i2 ω t
ω
d e+i2 ω t

0
0

⎤

⎥
⎥
−i e−i2 ω t ⎥
⎥
⎥.
⎥
0
⎥
d e−i2 ω t ⎦
−ω

(29)

Then, the fundamental equation (27) is written as follows⎡

ω⎢ d e+i2 ω t⎢⎢di ψ̂(t) = ⎢⎢⎢ 00
dt
⎢
⎣
0
0

d e−i2 ω t
−ω
0
0
0
0

0
+i2 ω tieωd e+i2 ω t
00−i e−i2 ω t
0d e−i2 ω t
−ω
0
0

0
0
0
i e+i2 ω t
ω
d e+i2 ω t

0
0

⎤

⎥
⎥
−i e−i2 ω t ⎥⎥⎥ ψ̂(t).
⎥0⎥−i2 ω t ⎦
de
−ω

(30)

The coupled state of the oscillation mode corresponding to (30) is shown in Fig. 2.
Let us consider the solution of the degenerate modes of the fundamental equation (30). First, we
introduce several functions δ1↑ (t), δ1↓ (t), δ2↑ (t), δ2↓ (t), δ3↑ (t), δ3↓ (t) that represent the deviation from theeigenfrequency ∓ω of the symmetrizable fundamental equations. The solutions to the fundamental
equations take the following form↑↑

ψ1↑ (t) = e−iθ1 (t) = e−iω t+iδ1 (t) ,

(31)

↓
↓
ψ1↓ (t) = e+iθ1 (t) = e+iω t+iδ1 (t) .

(32)

↑
↑
ψ2↑ (t) = e−iθ2 (t) = e−iω t+iδ2 (t) ,

(33)

164

Fig. 2.

Coupling states of 6 oscillation modes.
↓↓ψ2↓ (t) = e+iθ2 (t) = e+iω t+iδ2 (t) .
−iθ3↑ (t)ψ3↑ (t) = e+iθ3↓ (t)ψ3↓ (t) = e=e=e−iω t+iδ3↑ (t)+iω t+iδ3↓ (t)

(34)

,

(35)

.

(36)

Substituting (31)–(36) into the fundamental equation (30), we obtain the time evolution equations
for the phases. Taking the time evolution equation for δ1↑ (t) and δ1↓ (t) as an example, we get↓↑↓↑d δ1↑ (t)= −d e+i (δ1 (t)−δ1 (t)) + i e+i (δ2 (t)−δ1 (t)) ,
dt
↑
↓
↑
↓
d δ1↓ (t)= −d e+i (δ1 (t)−δ1 (t)) − i e+i (δ2 (t)−δ1 (t)) .
dt

(37)
(38)

Because the phase diﬀerence is not always a real number, δ1↑ (t) and δ1↓ (t) are written as complex
numbers as follows:δ1↑ (t) = Re[δ1↑ (t)] + i Im[δ1↑ (t)],δ1↓ (t) = Re[δ1↓ (t)] + i Im[δ1↓ (t)].

(39)

Substituting (39) into (37) and (38) gives the following time evolution equations for the real andimaginary parts of the phase diﬀerence.
d Re[δ1↑ (t)]
=
dt


↓
↑
π
+ d e−Im[δ1 (t)]+Im[δ1 (t)] sin Re[δ1↓ (t)] − Re[δ1↑ (t)] −2↓↑−Im[δ2↓ (t)]+Im[δ1↑ (t)]+esin Re[δ2 (t)] − Re[δ1 (t)] − π ,(40)d Re[δ1↓ (t)]
=
dt


↓
↑
π
+ d e+Im[δ1 (t)]−Im[δ1 (t)] sin Re[δ1↑ (t)] − Re[δ1↓ (t)] −
2


↓
↑
+ e+Im[δ1 (t)]−Im[δ2 (t)] sin Re[δ2↑ (t)] − Re[δ1↓ (t)] ,(41)

d Im[δ1↑ (t)]
=
dt



↓
↑
+ d e−Im[δ1 (t)]+Im[δ1 (t)] sin Re[δ1↓ (t)] − Re[δ1↑ (t)] + π↓↑π+ e−Im[δ2 (t)]+Im[δ1 (t)] sin Re[δ2↓ (t)] − Re[δ1↑ (t)] +,2d Im[δ1↓ (t)]
=
dt


↓

(42)

↑

+ d e+Im[δ1 (t)]−Im[δ1 (t)] sin Re[δ1↑ (t)] − Re[δ1↓ (t)] − π↓↑π+ e+Im[δ1 (t)]−Im[δ2 (t)] sin Re[δ2↑ (t)] − Re[δ1↓ (t)] −
.
2

(43)

The time evolution equations (40) and (41) of the real parts of the phase diﬀerence are similar to
the model of phase synchronization called the Kuramoto model [34] or the Sakaguchi model [35]. We

165

call these synchronization models in this paper the Kuramoto model. The signiﬁcant diﬀerence fromthe Kuramoto model lies in the coeﬃcients of the sine function in (40) and (41). In our model, thecoeﬃcients of the sine function are not constant. However, when the exponential function, which isa coeﬃcient of the sine function, is considered to be suﬃciently large, phase synchronization occurs,
as in the Kuramoto model, where the argument of the sine function becomes zero. Since there are
two or more diﬀerent terms of the sinusoidal function, such as (40) and (41), we should conﬁrm
if the corresponding phase synchronizations are compatible or not. Since phase synchronization isincompatible only if one of the terms becomes dominant, Kuramoto-model-like phase synchronization
occurs and the phase diﬀerence is ﬁxed at a speciﬁc value.
On the other hand, the time evolution equations (42) and (43) for the imaginary part do not trigger
phase synchronization, but they are inﬂuenced by the diﬀerence of real parts of phases. If some ofthe phase diﬀerences of real parts of phases are ﬁxed, the sign of the sine functions of the ﬁrst andsecond terms on the right-hand side are ﬁxed. Then, the rate of change of the imaginary part of thephase becomes ﬁxed at either positive or negative values.
Substituting (39) into (31) and (32), ψ1↑ (t) and ψ1↓ (t) are written as follows↑↑

ψ1↑ (t) = e−Im[δ1 (t)] e−i ω t+i Re[δ1 (t)] ,−Im[δ1↓ (t)]ψ1↓ (t) = ee+i ω t+i Re[δ1↓ (t)]

(44)

.

(45)
↑

↓

In (44), (45), as the phase synchronization of Re[δ1↑ (t)], Re[δ1↓ (t)] progresses, e−Im[δ1 (t)] , e−Im[δ1 (t)]increases exponentially with time. Therefore if Im[δ1↑ (t)] or Im[δ1↓ (t)] approaches −∞, the amplitudeof ψ1↑ (t) or ψ1↓ (t) diverges. This means oscillation energy divergence, which is one model that can
describe ﬂaming phenomena. This result shows that divergence might occur even if eigenvalues arereal numbers given that the eigenvalues are degenerate.

3.3 Fundamental equation of user dynamics with degeneration of eigenvalue 0
As mentioned above, the unitary-transformed fundamental equation (27) does not necessarily represent the dynamics of actual networks, and therefore might not lead directly to ﬂaming phenomena
in real networks. The dynamics of actual networks are described by the fundamental equation (25),
to which the unitary transformation is not applied. The extended fundamental equation (25) for the
network model shown in Fig. 1 is written as follows:
⎡

ω
⎢ d
⎢
⎢
d
i ψ̂(t) = ⎢⎢⎢ 00
dt
⎢
⎣ 0
0

d
−ω
0
0
0
0

0
i
ω
d
0
0

−i
0
d
−ω
0
0

0
0
0
i
ω
d

⎤
0
0 ⎥
⎥
−i ⎥
⎥
⎥ ψ̂(t).
0 ⎥
⎥
d ⎦
−ω

(46)

If we cast the solutions to the equation as (31)–(36), and then enter them into (46), we can derive
the time evolution equations for the phases by a procedure similar to that shown in the previous
subsection. The time evolution equations for δ1↑ (t) and δ1↓ (t) are written as follows:



↓
↑
d Re[δ1↑ (t)]= +d e−Im[δ1 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ1↓ (t)] − π/2 − Re[δ1↑ (t)]dt+ sin(2 ω t) sin Re[δ1↓ (t)] − Re[δ1↑ (t)]



↓
↑
+ e−Im[δ2 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ2↓ (t)] + π − Re[δ1↑ (t)]+ sin(2 ω t) sin Re[δ2↓ (t)] − π/2 − Re[δ1↑ (t)] ,




↑
↓
d Re[δ1↓ (t)]= +d e−Im[δ1 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ1↑ (t)] − π/2 − Re[δ1↓ (t)]dt+ sin(2 ω t) sin Re[δ1↑ (t)] + π − Re[δ1↓ (t)]



↑
↓
+ e−Im[δ2 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ2↑ (t)] − Re[δ1↓ (t)]

166

(47)



+ sin(2 ω t) cos Re[δ2↑ (t)] − π/2 − Re[δ1↓ (t)] ,

(48)




↓
↑
d Im[δ1↑ (t)]= +d e−Im[δ1 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ1↓ (t)] + π − Re[δ1↑ (t)]dt+ sin(2 ω t) sin Re[δ1↓ (t)] − π/2 − Re[δ1↑ (t)]



↓
↑
+ e−Im[δ2 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ2↓ (t)] + π/2 − Re[δ1↑ (t)]+ sin(2 ω t) sin Re[δ2↓ (t)] + π − Re[δ1↑ (t)] ,

(49)




↑
↓
d Im[δ1↓ (t)]= +d e−Im[δ1 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ1↑ (t)] + π − Re[δ1↓ (t)]dt+ sin(2 ω t) cos Re[δ1↑ (t)] + π/2 − Re[δ1↓ (t)]



↑
↓
+ e−Im[δ2 (t)]+Im[δ1 (t)] cos(2 ω t) sin Re[δ2↑ (t)] − π/2 − Re[δ1↓ (t)]+ sin(2 ω t) sin Re[δ2↑ (t)] + π − Re[δ1↓ (t)] .

(50)

In (47)–(50), unlike (40)–(43), the coeﬃcients of sin(2 ω t) or cos(2 ω t) appear in each term, making
it diﬃcult for Kuramoto-model-like phase synchronization to occur.
However, there is an exceptional situation wherein the unitary-transformed fundamental equation (30) of the networks matches the fundamental equation (46) of the real network. It is when thedegenerate eigenvalue is zero as follows:
⎡

0
⎢ d
⎢
⎢
d
i ψ̂(t) = ⎢⎢⎢ 00
dt
⎢
⎣ 0
0

d
0
0
0
0
0

⎤
0 −i 0 0
i 0 0 0 ⎥⎥0 d 0 −i ⎥⎥⎥ ψ̂(t).
d 0 i 0 ⎥⎥0 0 0 d ⎦0 0 d 0(51)This is equivalent to the case of ω = 0 in the unitary-transformed fundamental equation (30), whichsuggests that ﬂaming phenomena may occur.
Note that the matrix that appeared in (51) has only one eigenvalue of 0. The matrix appears to be
similar in the Jordan canonical form, but it is not. The non-diagonal component causes coupling of theseparated subnetworks by weak ties (see Appendix). Thus, the situation where eigenvalue 0 couples
as a Jordan block corresponds to the situation where coupling occurs in the divided communities
shown on the left of Fig. 3 which resolves the network fragmentation, as shown on the right of Fig. 3.
Since we are considering a limited change in which only the oscillation modes with eigenvalues of 0
couple; this represents the situation in which the divided communities are weakly coupled with eachother.

Fig. 3.

Network structure when oscillation mode 0 instances are coupled.

167

4. Numerical experiments
4.1 Outline of numerical experiments
The previous section considered the fundamental equations of user dynamics with degenerate eigenvalues and showed that ﬂaming phenomena might occur even when all eigenvalues are real numbers
if eigenvalue degeneration occurs with eigenvalue 0. However, since several parameters are involvedin the occurrence of ﬂaming phenomena, we have to conﬁrm whether ﬂaming phenomena can actually occur. In this section, we use numerical experiments to conﬁrm that the ﬂaming phenomena
associated with the degeneration of eigenvalue 0 can actually occur.
The target of the numerical experiments is the system described by the fundamental equation (51).
This is the situation described in Sec. 3.2, where the three divided communities shown in Fig. 3 haveweak ties and coupling between the three zero eigenvalues, represented by the Jordan canonical form
in appearance, has occurred. Given the initial conditions shown in Table I, we evaluate the time
evolution of the phase of the solution. The time evolution of the phase of the solution is analyzed
separately for real and imaginary parts, and each time evolution equation is evaluated from thefollowing perspectives.
• For the time evolution of the real part of the phase, we investigate whether the phase diﬀerence
is ﬁxed due to Kuramoto-like synchronization or not. If the phase diﬀerence is not ﬁxed, we
also investigate whether the phase diﬀerence is limited to a speciﬁc range.
• For the time evolution of the imaginary part of the phase, we investigate the inﬂuence of the
behavior of the phase diﬀerence obtained from the time evolution of the real part of the phase.
Table I.

ParameterdRe[δ1↑ (0)]Re[δ1↓ (0)]Re[δ2↑ (0)]Re[δ2↓ (0)]Re[δ3↑ (0)]Re[δ3↓ (0)]Initial condition.

Value
1
0
0
0
0
0
0

ParameterωIm[δ1↑ (0)]Im[δ1↓ (0)]Im[δ2↑ (0)]Im[δ2↓ (0)]Im[δ3↑ (0)]Im[δ3↓ (0)]

Value
0
0
0
0
0
0
0

4.2 Results
Figure 4 shows the phase diﬀerence determined from the time evolution equation of the real part ofthe phase. The horizontal axis shows the elapsed time, and the vertical axes in (a), (b), (c), (d), (e)and (f) plot Re[δ1↓ ] − Re[δ1↑ ], Re[δ2↓ ] − Re[δ1↑ ], Re[δ2↑ ] − Re[δ1↓ ], Re[δ2↓ ] − Re[δ2↑ ], Re[δ3↓ ] − Re[δ2↑ ] andRe[δ3↑ ] − Re[δ3↓ ], respectively. Comparing the phase diﬀerences in Fig. 4, we can classify them into two
types: phase diﬀerence 4(a), in which the oscillations are damped and converge to a speciﬁc value,and phase diﬀerence 4(b), 4(c), 4(d), 4(e), and 4(f), in which the oscillations are not damped and do
not converge.
First, we analyze from the phase diﬀerences associated with the time evolution equation (40), i.e.,
Figs. 4(a) and 4(b). If the sine functions in the right-hand side of this equation cause Kuramoto-likesynchronization, the ﬁrst term on the right-hand side converges onπRe[δ1↓ ] − Re[δ1↑ ] = + .
2
Similarly, the second term on the right-hand side converges onRe[δ2↓ ] − Re[δ1↑ ] = +π.
Figure 4(a) shows the phase diﬀerence converging on 0, which is diﬀerent from the value “+π/2”
predicted by the Kuramoto model. In addition, Fig. 4(b) shows that the phase diﬀerence does notconverge.

168

Fig. 4.

Time evolution of the diﬀerence of real parts of phases.

Next, we examine the phase diﬀerences associated with the time evolution equation (41), i.e.,
Figs. 4(a) and 4(c). If the sine functions on the right-hand side of this equation cause Kuramoto-likesynchronization, the ﬁrst term on the right-hand side converges onπRe[δ1↑ ] − Re[δ1↓ ] = −(Re[δ1↓ ] − Re[δ1↑ ]) = + .
2
Similarly, the second term on the right-hand side converges onRe[δ2↑ ] − Re[δ1↓ ] = 0.
Figure 4(a) shows that the phase diﬀerence converges on 0, which is diﬀerent from the value “−π/2”
predicted by the Kuramoto model. In addition, Fig. 4(c) shows that the phase diﬀerence does notconverge. The reason why the synchronization by the Kuramoto model does not occur is that thesine functions of the ﬁrst terms on the right-hand side of (40) and (41) disturb phase synchronization.
Note that, if Kuramoto-like synchronization does occur, Eqs. (42) and (43) show that the amplitude
increases exponentially, which alters the evolution of the imaginary part of the phase. Although there
is no Kuramoto-like synchronization here, it is necessary to investigate how the amplitude is aﬀected
through the behavior of the imaginary part of the phase.
Since the coeﬃcient of the sine functions on the right-hand sides of (42) and (43) is always positive,
the sign of the sine functions determines whether the imaginary part of the phase increases or decreases
with time. Therefore, we examine the sign of the sine functions in (42) and (43) using the diﬀerenceof the real part of the phase as shown in Fig. 4.
First, we examine the sign of the sine function on the right-hand side of (42). The phase diﬀerences
contributing to the sine function in the right-hand side of (42) are (a) and (b) in Fig. 4. In Fig. 4(a),the phase diﬀerence in the real part converges on Re[δ1↓ ] − Re[δ1↑ ] → 0 with damped oscillations. In
this case, the sine function of the ﬁrst term on the right-hand side of the corresponding equation (42)can be written as
sin Re[δ1↓ ] − Re[δ1↑ ] + π → sin π = 0.
Therefore, the sine function of the ﬁrst term on the right-hand side of (42) does not contribute to theincrease or decrease of Im[δ1↑ (t)]. On the other hand, in Fig. 4(b), the phase diﬀerence of the real partdoes not converge but oscillates within a certain range over time. Note that the minimum value is

169


 π
min Re[δ2↓ (t)] − Re[δ1↑ (t)] < ,2

but is very close to π/2. Details about how “close” will be given later. The maximum value is 4π
max Re[δ2↓ (t)] − Re[δ1↑ (t)] .
5Then, the sine function of the second term on the right-hand side of the corresponding equation (42)can be written as


 π


 π
> 0, sin max Re[δ2↓ (t)] − Re[δ1↑ (t)] +< 0.
sin min Re[δ2↓ (t)] − Re[δ1↑ (t)] +2
2
Therefore, the sign of the sine function changes with time. As a result, the temporal change inIm[δ1↑ (t)] shows a repeated increase and decrease behavior cycle.
Next, we investigate the sign of the sine functions on the right-hand side of (43). The phase
diﬀerences contributing to the sine functions on the right-hand side of (43) are (a) and (c) in Fig. 4.
Figure 4(a) shows a damped oscillation that converges to Re[δ1↑ ] − Re[δ1↓ ] → 0. In this case, the ﬁrst
term of the right-hand side the corresponding equation, (43), can be written as
sin Re[δ1↑ ] − Re[δ1↓ ] − π → sin(−π) = 0.
Therefore, the sine function of the ﬁrst term of the right-hand side of (43) does not contribute to theincrease or decrease of Im[δ1↓ (t)]. On the other hand, in Fig. 4(c), the phase diﬀerence of the real partdoes not converge, but oscillates within a certain range over time. The range with minimum value
close to π/5 is written as follows
 πmin Re[δ2↑ (t)] − Re[δ1↓ (t)]  .
5
The maximum value is written as follows πmax Re[δ2↑ (t)] − Re[δ1↓ (t)] < .
2Then, the range of values of the sine function in the second term of the right-hand side of thecorresponding equation, (43), is always negative, that is π
sin Re[δ2↓ (t)] − Re[δ1↑ (t)] −< 0.
2Thus, although the right-hand side of (43) diﬀers from the phase synchronization of the Kuramotomodel, the second term on the right-hand side of (43) produces a tendency for Im[δ1↓ (t)] to decrease,since the second term is always negative.
Figure 5 shows the time evolution for the imaginary part of the phases. In Figs. 5, (a), (b), (c), (d),(e) and (f) represent Im[δ1↑ (t)], Im[δ1↓ (t)], Im[δ2↑ (t)], Im[δ2↓ (t)], Im[δ3↑ (t)], and Im[δ1↓ (t)], respectively.
Comparing the phase diﬀerences in Fig. 5, we can classify them into two types; 5(a) and 5(b), in
which the values tend to decrease, and 5(c), 5(d), 5(e), and 5(f), in which the values keep withincertain ranges. We want to investigate the existence of oscillation mode divergence, so we analyzeIm[δ1↑ (t)] and Im[δ1↓ (t)] in Figs. 5(a) and (b).
The graphs in (a) Im[δ1↑ (t)] and (b) Im[δ1↓ (t)] have two common features. The ﬁrst is the presenceof spikes that are periodic and whose height decays. The second is the gradual global decreasein Im[δ1↑ (t)] and Im[δ1↓ (t)]. The spikes occur in opposite phases in the two graphs, and the global
decreasing trend is almost identical in both graphs.
We analyze the occurrence of the spikes. The occurrence of spikes indicates that the sign of the rateof change of Im[δ1↑ (t)] and Im[δ1↓ (t)] changes. In the right-hand side of (42) and (43), there are two
factors that cause the sign to change. The ﬁrst is the eﬀect of the ﬁrst term. This eﬀect can be seenfrom Fig. 4(a), which represents the oscillations that decay in the opposite phase. The second is theeﬀect of the second term. However, in the second term on the right-hand side, only (42) changes sign,

170

Fig. 5.

Time evolution of the imaginary part of the phase.

(43) does not. Considering the fact that the spikes appear in both Figs. 5(a) and 5(b) in common,and that they have the characteristic of attenuating in the opposite phase, we can recognize that thisis the eﬀect of the ﬁrst term on the right-hand side.
Next, we analyze the nature of the gradual global decrease in Im[δ1↑ (t)] and Im[δ1↓ (t)] to negativevalues. The ﬁrst terms on the right-hand side of (42) and (43) do not contribute to the global characteristics of the phenomena, because they represent the property of damping with short periodicity.
Therefore, this can be due to the eﬀect of the second term on the right-hand side of (42) and (43).
While the sine function of the second right-hand side term in (42) has a sign change, the sine functionof the second right-hand side term in (43) has no sign change and always takes a negative value. As
can be seen from Fig. 4(b), the sine function of the second term on the right-hand side in (42) is very
close to 0, even if a sign change occurs. Thus, the fact that Figs. 5(a) and (b) globally behave insimilar fashion means that the sine function of the second term on the right-hand side in (43) is not a
factor that causes a signiﬁcant diﬀerence. Next, we focus on the coeﬃcients before the sine function↓↑in (42) and (43). The coeﬃcients of the sine function in (42) and (43) are e−Im[δ2 (t)]+Im[δ1 (t)] and↓↑e−Im[δ1 (t)]+Im[δ2 (t)] . If Im[δ1↑ (t)] and Im[δ1↓ (t)] continue to decrease at negative values, the eﬀect of thesecond term on the right-hand side weakens. Therefore, we consider that the coeﬃcients in Figs. 5(a)and (b) contribute to the gradual decrease in the global scale.
The observation drawn from Figs. 5(a) and (b) does not clarify whether the decrease of the imaginary
part of the phase is bounded or not. Therefore, we further investigate the characteristics of the time
variation of Im[δ1↑ (t)] and Im[δ1↓ (t)] in more detail. Figures 6(a) and (b) show the time evolution(logarithmic scale) of |Im[δ1↑ (t)]| and |Im[δ1↓ (t)]|. We conﬁrm from Figs. 6(a) and (b) that |Im[δ1↑ (t)]|and |Im[δ1↓ (t)]| increase linearly, i.e., the absolute values of the imaginary parts of the two phases
diverge logarithmically with time. Therefore, we can conclude that Im[δ1↑ (t)] and Im[δ1↓ (t)] diverge,
taking negative values.
By substituting the divergence of the imaginary components of the phase shown in Figs. 6(a) and(b) into (44) and (45), we can see that the state of the oscillation mode diverges. From the above
result, the logarithmic divergence of the imaginary part of the phases causes divergence in the strengthof user dynamics to occur when divided communities experience correlation.

171

Fig. 6. Time evolution of the imaginary part of phase with respect to thetime (logarithmic scale).

5. Conclusions
In this paper, to illustrate the mechanism of ﬂaming phenomena in more detail, we investigated thecharacteristics of the user dynamics generated by the interaction of network oscillation modes, which
are degenerate with eigenvalue 0 and inﬂuence each other in Jordan canonical form. We showed
theoretically that the fundamental equations of the real network of degenerate modes with eigenvalue
0 are equal to the unitary-transformed fundamental equation of networks, and so can cause ﬂaming
phenomena. Numerical simulations demonstrated that divergence could occur in a model representing
the coupling of oscillation mode 0 instances.
The situation in which the eigenvalue 0 instances couple as a Jordan block corresponds to the situation in which a fragmented community is coupled and the fragmentation of the network is resolved.
In general, changes in the network structure also have a signiﬁcant impact on the graph spectrum,
but in this work we focus only on changes where the degeneration of eigenvalue 0 is resolved, thatis, we consider only limited changes in the network structure. Thus, even if coupling of fragmented
communities does occur, it is possible that we are considering only the case wherein the coupling isrelatively weak. Degenerate eigenvalues are extremely uncommon and unlikely to occur in OSNs, withthe exception of 0 eigenvalue degeneracy. This degeneracy actually occurs because of the phenomenon
related to the fragmentation of the network structure. As mentioned in the Introduction, weak ties
between fragmented communities that do not usually closely interact can cause signiﬁcant changes inuser dynamics. This is an interesting subject for further research. The mechanism of ﬂaming phenomena discussed in this paper corresponds to an example of user dynamics when weak connections
occur between divided communities, and it is an example of user dynamics related to “weak ties.”
Therefore, our results can be useful in modeling interesting user dynamics created by weak ties.

Acknowledgments
This research was supported by Grant-in-Aid for Scientiﬁc Research (B) No. 19H04096 (2019–2021)
and No. 20H04179 (2020–2022), and Grant-in-Aid for Scientiﬁc Research (C) No. 18K11271 (2018–2020) from the Japan Society for the Promotion of Science (JSPS).

Appendix
Let us consider the square of the following matrix:

172

⎡

0
⎢ 0
⎢
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢ 0
⎣
0

1
0
0
0
0

0 0
1 0
0 0
0 ω1
0 0

0 0

0

0
0
0
0
ω2
0

⎤2
⎡
0
0
⎥
⎢ 0
0 ⎥
⎢
⎥
⎢
0 ⎥
⎢ 0
⎥
⎢
0 ⎥ =⎢ 0
⎥
⎢
⎢ 0
0 ⎥
⎦
⎣
..
.
0

0
0
0
0
0

1 0
0 0
0 0
0 ω12
0 0

0 0

0

0
0
0
0
ω22
0

⎤
0
0 ⎥
⎥
0 ⎥
⎥
⎥
0 ⎥.
⎥
0 ⎥
⎦
..
.

By arranging rows and columns, we have the following matrix, which is the Jordan canonical form:
⎡

0
⎢ 0
⎢
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢ 0
⎣
0

1
0
0
0
0

0 0
0 0
0 0
0 ω12
0 0

0 0

0

0
0
0
0
ω22
0

⎤
0
0 ⎥
⎥
0 ⎥
⎥
⎥
0 ⎥.
⎥
0 ⎥
⎦
..
.

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