IEICE TRANS. COMMUN., VOL.E102–B, NO.8 AUGUST 2019

1554

PAPER

New Model of Flaming Phenomena in On-Line Social Networks
Caused by Degenerated Oscillation Modes
Takahiro KUBO†a) , Chisa TAKANO†† , and Masaki AIDA† , Members

SUMMARY The explosive dynamics present in on-line social networks,
typically represented by flaming phenomena, can have a serious impact on
not only the sustainable operation of information networks but also on
activities in the real world. In order to counter the flaming phenomenon, it
is necessary to understand the mechanism underlying the generation of theflaming phenomena within an engineering framework. This paper discusses
a new model of the generating mechanism of the flaming phenomena. Our
previous study has shown that the cause of flaming phenomena can, byreference to an oscillation model on networks, be understood complex
eigenvalues of the matrix formed to describe oscillating phenomena. In
this paper, we show that the flaming phenomena can occur due to coupling
between degenerated oscillation modes even if all the eigenvalues are real
numbers. In addition, we investigate the generation process of flaming
phenomena with respect to the initial phases of the degenerated oscillation
modes.
key words: Laplacian matrix, directed graph, flaming, Jordan canonical
form

1.

Introduction

Recent developments in social networking services (social
networks hereafter) have explosively activated informationexchange among users. Active information exchange createsa situation where the activities of individual users of on-line
social networks strongly influence each other. While such asituation has the positive effect of promoting the formationof active on-line communities of dynamic user, we recognizethat more attention must be paid to the generation of negative effects. The user dynamics in on-line social networks
can seriously impact on not only the sustainable operationof information networks, but also on activities in the realworld. Typical examples are flash crowds on web servers
and harmful rumors about disasters, respectively. Therefore,
understanding the user dynamics present in on-line social
networks is expected to yield important clues for resolving
both issues. In particular, the flaming phenomena is a typicalexample of the negative influence caused by user dynamics;
flaming refers the phenomena in which the strength of user
dynamics, such as number of comments on SNS posts, becomes imbalanced due to the mutual influence among users.
Since flaming phenomena arise frequently and are becoming
Manuscript received August 9, 2018.
Manuscript revised December 15, 2018.
Manuscript publicized January 24, 2019.
† The authors are with the Graduate School of Systems Design,
Tokyo Metropolitan University, Hino-shi, 191-0065 Japan.
†† The author is with the Graduate School of Information Sciences, Hiroshima City University, Hiroshima-shi, 731-3194 Japan.
a) E-mail: kubo-takahiro@ed.tmu.ac.jp
DOI: 10.1587/transcom.2018EBT0002

a significant social issue, it is important for the success of
on-line social networks to understand of mechanism underlying the generation of the flaming phenomena and establish
suitable countermeasure technologies [1].
This paper shows, from a theoretical point of view, thatthe flaming phenomena can occur due to coupling between
degenerated oscillation modes in the oscillation model on
networks even if all the eigenvalues are real numbers. Thisnew model of the flaming phenomena, which is based on
the coupling of oscillation modes, differs from the previous
flaming phenomena model in that it allows the eigenvaluesof the Laplacian matrix to be real numbers. To the best of
our knowledge, this model is completely original.
This paper is organized as follows. In Sect. 2, we introduce some related work. Section 3 provides the preliminaries
by briefly explaining the oscillation model on directed graphs
and the fundamental equation that describes the oscillation
model. In Sect. 4, we introduce and analyze the coupling effects caused by the degenerated oscillation modes. In Sect. 5,
we investigate the relationship between the initial phase of
the oscillation modes and occurrence of the flaming phenomena from the degenerated oscillation modes. Finally, we
conclude this paper in Sect. 6.
2.

Related WorkIn various networks in the real world, the state of a node in
networks is changed over time by the effect of other nodes
through network links. Therefore, in order to know the node
dynamics in networks, it is important to understand the effectof network links. Numerous studies dealing with network
dynamics have been published [2]–[4]. In [2], malware propagation is studied by using a two layer epidemic model: theupper layer focuses on large scale networks, such as domainsof the Internet, while the low layer model offers better accuracy than the single layer epidemic models used in malware
modeling. In [3], the dynamics of an epidemic model arestudied by using a susceptible-infected-susceptible epidemic
model that incorporates with multistage infection (infection
delay) and an infective medium (propagation vector) overcomplex networks, which typically exhibit nonlinear characteristics. In [4], consensus problems for networks with directed information flow are studied by analyzing three cases:
1) directed networks with fixed topology; 2) directed networks with switchable topology; and 3) undirected networks
with communication time-delays and fixed topology. However, there has never been a model capable of describing

Copyright © 2019 The Institute of Electronics, Information and Communication Engineers

KUBO et al.: NEW MODEL OF FLAMING PHENOMENA IN ON-LINE SOCIAL NETWORKS CAUSED BY DEGENERATED OSCILLATION MODES

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the explosive user dynamics like the flaming phenomena in
on-line social networks.
We have proposed an oscillation model on networks andclarified the divergence of the user state [1], [5], [6]. According to [7], [8], applying the oscillation model to networks cangive a generalized concept of node centrality [9], [10], whichindicates the strength of activity of network nodes. On thisbasis, we can define the flaming phenomena as a divergencein node centrality over an entire network [11]. In addition,
we clarified that the flaming phenomena are caused by the
Laplacian matrix representing the network structure having
complex eigenvalues, and that flaming phenomena do notoccur if the network structure has only eigenvalues of real
numbers [1], [11]. On the other hand, there is true thatsome Laplacian matrices describing social networks cannot
be diagonalized mathematically. That case yields dynamics
greatly different from the case in which the Laplacian matrix
can be diagonalized. In particular, the flaming phenomena
can occur if the Laplacian matrix can not be diagonalizedeven if the eigenvalues of the Laplacian matrix are real numbers. However, those dynamics have not been examined sofar.
The contribution of this study lies in dealing with a
network structure in which the Laplacian matrix can not be
diagonalized. In this study, we reveal a new mechanism thatyields the flaming phenomena even if all the eigenvalues of
the Laplacian matrix are real numbers.
3.

Preliminaries

3.1

Oscillation Model on Directed Networks

On-line social networks must be modelled with directed
graphs to describe the strengths of the influence between
users, which are generally asymmetrical. Here, we introduce the oscillation model on on-line social networks [1].
The oscillation model on networks can simply describe
the situation in which users influence each other. As thismodel, we consider the simplest and universal model, called
the minimal model; it provides a simple expression of userstates and universal interaction between users. To indicate
the state of node i at time t, we assume that user state is
written as the one-dimensional variable x i (t), where i =1, 2, . . . , n and n is the number of users. In addition, we
define the state vector for all users asx(t) := t(x 1 (t), . . . , x n (t)).

(1)

The interaction between users is described as follows. Useri receives the force from neighboring node j such that thedifference |x i (t) − x j (t)| becomes small. The strength of theforce is proportional to the difference x i (t) − x j (t) as−wi j (x i (t) − x j (t)),(2)

where wi j > 0 is a constant. Since the interaction between
users is asymmetrical, wi j , w ji in general. Then, theequation of motion of the state vector of users is written asd2x(t) = −L x(t),
dt 2

(3)

where L is the Laplacian matrix [12] of the directed graphwhose weight of directed link (i → j) is given by wi j . We canalso consider the additional damping effect in the equationof motion. However, in order to focus on understanding
the mechanism of generating flaming phenomena, this paper
does not take the damping effect into consideration. We
rewrite the equation of motion by expressing the Laplacianmatrix in Jordan canonical form. Let the Jordan canonicalform of L be Λ, By using the square matrix P = [v1, . . . , vn ]where vi (i = 1, . . . , m, . . . , n) is the generalized eigenvectorof the Laplacian matrix L, the Jordan canonical form of Λ
is given byΛ = P−1 L P.

(4)

Then, we can write the equation of motion (3) as
d2y(t) = −Λ y(t),
dt 2

(5)

where y(t) = P−1 x(t).
If Λ is diagonalizable, (5) is decomposed into the equation of motion of n independent harmonic oscillators asy(t) = exp[∓i Λ1/2 t] y(0).

(6)

On the oscillation model for non-diagonalizable Λ, the equation of motion cannot be decomposed into independent harmonic oscillators due to the influence of the off-diagonalcomponents of Λ. To explicitly understand the influence ofthe off-diagonal components, we use the fundamental equation [1]. First, we introduce an n × n matrix H that satisfies
H 2 = Λ.

(7)

Then, the fundamental equation of (5) is written as±id φ(t)= H φ(t).
dt(8)

The solution, φ(t), of the fundamental equation (8) satisfiesd2dφ(t) = ∓i H φ(t) = −H 2 φ(t),2dt
dt
the solution of (8) is also the solution of the original equationof motion (5). Although the method of determining H from
(7) is not unique, we can choose any H that satisfies (7).
Note that the oscillation energy that is obtained from (6)gives the generalized notion of node centrality that includes
the conventional degree centrality and the betweenness centrality [7], [8]. The node centrality expresses the strengthof the activity of each node in the network. The oscillation
model on directed graphs posits that no divergence in theoscillation energy occurs when the Laplacian matrix L is
diagonalizable and all the eigenvalues are real numbers. In
other words, the diverge of the oscillation energy occurs only
when at least one pair of eigenvalues are complex numbers

IEICE TRANS. COMMUN., VOL.E102–B, NO.8 AUGUST 20191556

(they are complex conjugates). These diverging phenomenaof the oscillation energy have been studied as the flaming
phenomena on networks [1], [11].
3.2Unitary Transformation of Fundamental Equation

If Λ is non-diagonalizable, H is also non-diagonalizable.
However, even if Λ is in Jordan canonical form, H satisfying(7) does not have Jordan canonical form, in general. Here, let
us consider which of Λ and H provides the most fundamental
understanding. From the viewpoint of analyzing the users’
dynamics in networks associated with non-diagonalizable
Laplacian matrices, since the fundamental equation (8) is
described by H, the linear operator H is more fundamental
than Λ. On the other hand, since Λ is given by the elementary transformation of the Laplacian matrix, it leads toa concrete image of social network structures. In this paper, we adopt H as being fundamental for ease of analysis.
Hereafter, we introduce Ω that corresponds to H expressedin Jordan canonical form and we give Ω as the starting point
for analyzing the oscillation model on networks. Note that
this means Ω2 is not in Jordan canonical form, in general.
The attempt to derive a simple relationship between H andΛ (or Ω and Ω2 ) is discussed in Sect. 3.3.
We consider user dynamics in networks under the condition that the linear operator H is non-diagonalizable andits Jordan canonical form Ω is expressed as
ω1





Ω = 







1
ω1

..

.

..

.

0
1
ω1

0

ω2

1
..

.

1
ω2

..

.







 .








where Ω0J := diag(ω, . . . , ω) and ΩIJ is the remaining nondiagonal matrix. If we choose the ω as ω = ωi , it is possiblethat the diagonal component of ΩIJ is zero at the decomposition of ΩJ . However, in general, the diagonal component ofΩIJ is not zero. Let ψ(t) be the solution of the fundamental
equation associated with the linear operator ΩJ for the selected Jordan block. That is, we can write the fundamental
equation and its solution ψ(t) as
±i

1
ωi
0
0

0
..
.
..
0

.

0 

0 
 ,
1 
ωi 

d Ψ0 (t)= Ω0J Ψ0 (t),dtd ψI (t) ±i= Ψ0 (−t) ΩIJ Ψ0 (t) ψI (t),
dt

±i

(13)

(14)
(15)

where Ψ0 (t) is a matrix with the initial condition of Ψ0 (0) =I . The solution of (14) is given byΨ0 (t) = exp ∓i Ω0J t .
The initial condition of (13) is equal to the initial conditionof (15) as inψ(0) = Ψ0 (0) ψI (0) = I ψI (0) = ψI (0).

(16)

Note that Ψ0 (t) is proportional to the identity matrix
since the corresponding eigenvalues are degenerate. Therefore, Ψ0 (t) is commutable with any m × m matrix. By usingΩIJ Ψ0 (t) = Ψ0 (t) ΩIJ , we have
ΩIJ = Ψ0 (t) ΩIJ Ψ0 (−t),ΩJ· = Ψ0 (t) ΩJ· Ψ0 (−t).

(17)
(18)

Using the above relations, we can rewrite (15) as
(9)

±i


d ψI (t) = Ψ0 (−t) ΩIJ Ψ0 (t) ψI (t)dt= ΩIJ ψI (t),(19)

and we can also rewrite the fundamental equation (12) as(10)

where ωi is the diagonal component of i-th Jordan block andis a degenerated eigenvalue. In addition, m is the number of
duplicates of ωi .
Next, we decompose ΩJ into a certain matrix that is
proportional to the identity matrix and the other remaining
matrix asΩJ = Ω0J + ΩIJ,

(12)

where Ψ0 (t) and ψI (t) satisfy the following two equations
related to Ω0J and ΩIJ as

We focus on a Jordan block (m × m matrix) selected fromthe Jordan canonical form as
ωi

0
ΩJ =  00d ψ(t)= (Ω0J + ΩIJ ) ψ(t),dt
ψ(t) = Ψ0 (t) ψI (t),

(11)

±i


d ψ(t)  0= ΩJ + ΩIJ ψ(t)dt= Ω0J + Ψ0 (t) ΩIJ Ψ0 (−t) ψ(t).

(20)

This means the fundamental equation is invariant under theunitary transformation of ΩJ· as inΩJ· → Ψ0 (t) ΩJ· Ψ0 (−t).
The reason for choosing the above unitary transformationis convenience in the evaluations as demonstrated in thefollowing sections.

KUBO et al.: NEW MODEL OF FLAMING PHENOMENA IN ON-LINE SOCIAL NETWORKS CAUSED BY DEGENERATED OSCILLATION MODES

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σi σ j = −σ j σi,
3.3

Simple Correspondence between Linear Operator ΩJ
and the Network Structure

The linear operator ΩJ in fundamental equation (12) is notuniquely determined. In other words, we can select any ΩJas long as H satisfies (7). When utilizing this degree of
freedom to choose ΩJ , it is desirable to be able to makean explicit relationship between ΩJ and the structure of theoriginal network.
First of all, we decompose non-diagonalizable matrixΩIJ into diagonal matrix ΩIJ (d) and remainder matrix ΩIJ (a)which has only off-diagonal components, as inΩJ = Ω0J + ΩIJ = Ω0J + ΩIJ (d) + ΩIJ (a) .

where E is the 2 × 2 identity matrix. By using the Pauli matrices, we convert the fundamental equation (12) as follows±i0Ω̂J := Ω0J ⊗ σ3,
I (d)

Ω̂J

:= ΩIJ (d) ⊗ σ1,

(26)

I (a)
Ω̂J := ΩIJ (a) ⊗ σ2 .

In (26), ⊗ represents the Kronecker product. Ω̂J is a 2m ×2msquare matrix and ψ̂(t) is a 2m-dimensional vector↑↓(t), ψm(t)).
ψ̂(t) = t(ψ1↑ (t), ψ1↓ (t), ψ2↑ (t), ψ2↓ (t), . . . , ψm

+ Ω0J ΩIJ (d) + ΩIJ (d) Ω0J+ Ω0J ΩJI (a) + ΩIJ (a) Ω0J(22)

To simplify the calculation of (25), let us start fromthe unitary transformed version of the fundamental equation(20). The corresponding fundamental equation is written as±i

If the following anti-commutation relationsΩ0J ΩIJ (d) = −ΩIJ (d) Ω0JΩ0J ΩJI (a) = −ΩIJ (a) Ω0J

are established, all the cross terms in (22) can be canceled,
and we can obtain the simple relation of  2  I (d)  2  I (a)  2Ω0J + ΩJ
+ ΩJ
= Λ0 + ΛI = Λ,
(23)
where Λ0 is the diagonal matrix that satisfies Λ0 = (Ω0J ) 2 .
That is, the network structure of the non-diagonalizable matrix is written as

2 
2
ΛI = ΩIJ (d) + ΩIJ (a) .
(24)
The relation (24) can explicitly represent the correspondence
between the Laplacian matrix and the non-diagonalizable
part of Ω. If Ω is in Jordan canonical form, Λ simultaneously
has Jordan canonical form.
The above anti-commutation relations are not true forthe original matrices. However, by using the tensor product,
we can introduce the anti-commutation relation to our model.
We introduce the algebraic structure of the special unitarygroup SU(2) by using Pauli matrices [13], [14]:
"
#
"
#
"
#
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
relations:0d ψ̂(t)  0 ±i Ω̂0J t  I (d)I (a)= Ω̂J +eΩ̂J + Ω̂Je∓i Ω̂J t ψ̂(t).
dt(27)

The solution is given byψ̂(t) = Ψ̂0 (t) ψ̂I (t),ΩIJ (d) ΩIJ (a) = −ΩIJ (a) ΩIJ (d)σi2 = E,d ψ̂(t)
I (a)
I (d)
0
ψ̂(t),= Ω̂J ψ̂(t) = Ω̂J + Ω̂J + Ω̂J
dt
(25)

where

(21)

The square of ΩJ is written as 2 
2 
2Ω2J = Ω0J + ΩIJ (d) + ΩIJ (a)+ ΩIJ (d) ΩIJ (a) + ΩIJ (a) ΩIJ (d) .

(i , j).

(28)

where Ψ̂0 (t) is a 2m × 2m diagonal matrix and ψ̂I (t) is a
2m-dimensional vector that satisfy, respectively,±id Ψ̂0 (t)0= Ω̂J Ψ̂0 (t),dt±id ψ̂I (t)I= Ω̂J ψ̂I (t). (29)dt

0Since Ψ̂0 (t) = exp[∓i Ω̂J t], the solution of (27) is expressedas0ψ̂(t) = exp(∓i Ω̂J t) ψ̂I (t).

(30)

4.

Coupling Effects Caused by Degenerated Oscillation
Modes

4.1

Case Study of Three Degenerated Modes

Let us consider the simplest example of the degenerated
oscillation modes. For example, a network structure that can
exhibit triple degeneracy is the following. We provide below
an example of the Laplacian matrix that encompasses real
eigenvectors and degenerated oscillation modes.
 7

 −2
 −2
 −2
−9/4

−5
7
−3
−3
−9/4

−1
−3
9
−1
−9/4

0
−1
−3
7
−9/4

−1

−1
−1 .
−1

9 

(31)

IEICE TRANS. COMMUN., VOL.E102–B, NO.8 AUGUST 2019

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i

Fig. 1

d
ψ̂(t)
dt



ω
d e−i2 ω t
0−i e−i2 ω t
00
 d e+i2 ω t−ω
i e+i2 ω t
000
−i2 ω t−i2 ω t 
0
0
ω
d
e
0
−i
e
 ψ̂(t).
=  00d e+i2 ω t−ω
i e+i2 ω t0−i2 ω t 
0
0
0
0
ω
d
e


+i2
ω
t
0
0
0
0
de
−ω



Coupling of three pairs of oscillation modes.

(35)This Laplacian matrix can be transformed into Jordan canonical form as follows
0

0
0
0
0

0
9
0
0
0

0 0
0 

0 0
0 
10 1
0  .
0 10 1 
0 0 10(32)+iθ1↓ (t)Then, the linear operators of Ω̂J are written asω00Ω̂J = ΩJ ⊗ σ3 =  0
 0

0
ω
0
d
I (d)
Ω̂J = ΩJI (d) ⊗ σ1 =  0
 0
0
I (a)
I (a)
Ω̂J = ΩJ ⊗ σ2 = 0
0

+e


ω
 d e+i2 ω t=  00


0

0

d e−i2 ω t
−ω
0
0
0
0

0

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


0
0
0

−iθ2↑ (t)ψ2↑ (t) = e↑ei e+i2 ω t
ω
d e+i2 ω t

0
0




−i2
ω
t
−i e
 .
0

−i2
ω
t

de

−ω

(34)
By using (25), (26) and (34), the fundamental equation is
written as

(37)

,

(38)

=e

(39)
(40)
(41)

↓

↑

↓

↑

↑

↓

↑

↓

= ω + d e+i (δ1 (t)−δ1 (t)) − i e+i (δ2 (t)−δ1 (t)), (42)= ω − d e+i (δ1 (t)−δ1 (t)) − i e+i (δ2 (t)−δ1 (t)) . (43)Furthermore, by using the relationships of θ 1↑ (t) = ω t −δ1↑ (t)and θ 1↓ (t) = ω t + δ1↓ (t), we can write the temporal evolution
equation of δ1↑ (t), δ1↓ (t) asd δ1↑ (t)−i Ω0J ⊗ σ 3 t.

−iω t+iδ2↑ (t)=e

By substituting (36) and (37) into fundamental equation (35),
we have

dt
(33)

(36)

+iω t+iδ1↓ (t)↓↓ψ2↓ (t) = e+iθ2 (t) = e+iω t+iδ2 (t) .
↑↑ψ3↑ (t) = e−iθ3 (t) = e−iω t+iδ3 (t),↓↓ψ3↓ (t) = e+iθ3 (t) = e+iω t+iδ3 (t) .

dtd θ 1↓ (t)= Ω0J ⊗ σ3
ΩIJ (d) ⊗ σ1 + ΩIJ (a) ⊗ σ2

ψ1↓ (t) = ed θ 1↑ (t)0 0  ⊗ σ3,ω
0 0 
d 0  ⊗ σ1,
0 d 
1 0
0 1 ⊗ σ2,0 0

where ω + d is the eigenfrequency of the degenerated oscillation modes.
We adopt the unitary transformed version of fundamental equation (27). From (33), we introduce the 6 × 6 matrixΩ̂J by using the Kronecker product of the Pauli matrices as


0
0
0
I (d)
I (a)
Ω̂J + e+i Ω̂J t Ω̂J + Ω̂Je−i Ω̂J t↑ψ1↑ (t) = e−iθ1 (t) = e−iω t+iδ1 (t),There is coupling among the three degenerated modes. For
three degenerated modes selected from all modes of a directed graph, there are three pairs of oscillation modes (total
six modes). Each oscillation mode interacts the other asshown in Fig. 1. The three pairs of oscillation modes are
written asψ̂(t) = t ψ1↑ (t), ψ1↓ (t), ψ2↑ (t), ψ2↓ (t), ψ3↑ (t), ψ3↓ (t) .

+i Ω0J ⊗ σ 3 t

The fundamental equation considers all oscillation modes(35). First, we introduce the phase differences δ1↑ (t), δ1↓ (t),δ2↑ (t), δ2↓ (t), δ3↑ (t), δ3↓ (t); they represent the phases of theoscillation modes of angular frequency ∓ω. By using thephase difference, the solution of (35) is written asdt
d δ1↓ (t)
dt

↓

↑

↓

↑

↑

↓

↑

↓

= −d e+i (δ1 (t)−δ1 (t)) + i e+i (δ2 (t)−δ1 (t)),= +d e+i (δ1 (t)−δ1 (t)) + i e+i (δ2 (t)−δ1 (t)) .

(44)
(45)

(44) and (45) represent the differential equations of the phasecomponent of ψ1↑ (t), ψ1↓ (t). It is not always true that the phasedifferences are real numbers. Therefore, δ1↑ (t) and δ1↓ (t) are
written as
δ1↑ (t) = Re[δ1↑ (t)] + i Im[δ1↑ (t)],δ1↓ (t) = Re[δ1↓ (t)] + i Im[δ1↓ (t)].

(46)

By substituting (46) into equations (44) and (45), the temporal evolution equation of both the real and imaginary partsof δ1↑ (t) and δ1↓ (t) are written asd Re[δ1↑ (t)]dt=−Im[δ1↓ (t)]+Im[δ1↑ (t)]+desin(Re[δ1↓ (t)] − Re[δ1↑ (t)] −π)2

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↓

↑

+ e−Im[δ2 (t)]+Im[δ1 (t)] sin(Re[δ2↓ (t)] − Re[δ1↑ (t)] − π),(47)d Re[δ1↓ (t)]
dt
+de
+e

=

↑

+Im[δ1↓ (t)]−Im[δ1↑ (t)]+Im[δ1↓ (t)]−Im[δ2↑ (t)]sin(Re[δ2↑ (t)] − Re[δ1↓ (t)] − π),

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)] + ),2
(49)
d Im[δ1↓ (t)]
dt
+de

=

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

π
).
2
(50)

The temporal evolution equations of the real parts (47), (48)are similar to the mathematical model used to describe synchronization, i.e., the so-called Kuramoto model [15]. The
significant difference of (47), (48) from the Kuramoto model
lies in the coefficients of the sine functions; the coefficientsare not constant in our model. If the temporal change ofthe coefficients can be considered to be negligible, synchronization occurs also in our model. In this situation, theargument of the sine function becomes zero and phase synchronization occurs. If phase synchronization occurs in (47),Re[δ1↓ (t)] and Re[δ1↑ (t)] of the first term on the right-handside synchronize with the phase difference of π/2. Figure 2
shows the relationship of the phase differences of Re[δ1↑ (t)],Re[δ1↓ (t)], Re[δ2↑ (t)] and Re[δ2↓ (t)] in the parentheses of thesine functions in (47) and (48) after synchronization.
Unlike the case of real parts, the temporal evolution
equations of the imaginary parts (49) and (50) do not cause
phase synchronization. However, if Re[δ1↓ (t)] and Re[δ1↑ (t)]cause phase synchronization, both arguments of the sinefunctions in the first and second terms of the right handside of (49) and (50) become 3π/2. If d > 0, the change↑ψ1↑ (t) = e−Im[δ1 (t)] e−i ω t+i Re[δ1 (t)],πsin(Re[δ1↑ (t)] − Re[δ1↓ (t)]+ )2(48)d Im[δ1↑ (t)]rates of Im[δ1↑ (t)] and Im[δ1↓ (t)] are always negative becausesin(3π/2) = −1 on the right-hand side. By substituting theabove results into (36) and (37), ψ1↑ (t) and ψ1↓ (t) are writtenas−Im[δ1↓ (t)]ψ1↓ (t) = e+i ω t+i Re[δ1↓ (t)]

e

(51)

.

(52)

If Re[δ1↑ (t)] and Re[δ1↓ (t)] are synchronized, (51) and↑↓(52) increase because e−Im[δ1 (t)] and e−Im[δ1 (t)] increase autonomously with time. The divergence of those terms increases triggering divergence of the oscillation energy, andthis can be interpreted as a model that describes the flaming phenomena. The divergence of the oscillation energy
is exponential as is the divergence of the flaming phenomena if the eigenvalues of the Laplacian matrix are complex
numbers. However, there is a difference between the flaming
phenomena caused by the degenerated oscillation modes andthe flaming phenomena in the case that the eigenvalues of
the Laplacian matrix are complex numbers. The emergenceof the flaming phenomena caused by the degenerated oscillation modes can take time until divergence occurs because
the oscillation modes go through a process of synchronization before divergence occurs. Using the model analyzed inthis subsection, we confirmed that the flaming phenomena
can occur due to coupling between degenerated oscillation
modes even if all eigenvalues of H are real numbers. Thismeans that all eigenvalues of L are also real numbers.
4.2

Numerical Experiments

In this subsection, we show the divergence of the solutionsof the differential equations of the three pairs of degenerated oscillation modes to confirm the validity of the flaming
model discussed in Sect. 4.1. In a similar way to Eqs. (47),(48), (49), and (50), the temporal evolution equation of both
the real and imaginary parts of δ2↑ (t), δ2↓ (t), δ3↑ (t) and δ3↓ (t)are written asd Re[δ2↑ (t)]
dt

=

↓

↑

+ d e−Im[δ2 (t)]+Im[δ2 (t)] sin(Re[δ2↓ (t)] − Re[δ2↑ (t)] −
↓

π
)
2

↑

+ e−Im[δ3 (t)]+Im[δ2 (t)] sin(Re[δ3↓ (t)] − Re[δ2↑ (t)] − π),(53)d Re[δ2↓ (t)]dt=+Im[δ2↓ (t)]−Im[δ2↑ (t)]+de↓sin(Re[δ2↑ (t)] − Re[δ2↓ (t)] +

π
)
2

↑

+ e+Im[δ2 (t)]−Im[δ3 (t)] sin(Re[δ3↑ (t)] − Re[δ2↓ (t)] − π),(54)d Im[δ2↑ (t)]dtFig. 2

Relationship of phase differences after synchronization.

↓=↑+ d e−Im[δ2 (t)]+Im[δ2 (t)] sin(Re[δ2↓ (t)] − Re[δ2↑ (t)] + π)

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Table 1

Parameters of initial phase.

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

Value
1
0
0
0
0
0
0

Parameter

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

0
0
0
0
0
0

↑

+ e−Im[δ3 (t)]+Im[δ2 (t)] sin(Re[δ3↓ (t)] − Re[δ2↑ (t)] +d Im[δ2↓ (t)]dt=Fig. 3↓↓↑+ e+Im[δ2 (t)]−Im[δ3 (t)] sin(Re[δ3↑ (t)] − Re[δ2↓ (t)] +dt

Numerical results of Re[δ1↑ (t)], Re[δ1↓ (t)], Re[δ2↑ (t)], Re[δ2↓ (t)],Re[δ3↑ (t)] and Re[δ3↓ (t)].

↑+ d e+Im[δ2 (t)]−Im[δ2 (t)] sin(Re[δ2↑ (t)] − Re[δ2↓ (t)])

d Re[δ3↑ (t)]

π
),
2
(55)

π
),
2
(56)

=

↓
↑
π
+ d e−Im[δ3 (t)]+Im[δ3 (t)] sin(Re[δ3↓ (t)] − Re[δ3↑ (t)]− ),2(57)d Re[δ3↓ (t)]
dt
+de

=

+Im[δ3↓ (t)]−Im[δ3↑ (t)]d Im[δ3↑ (t)]dtπsin(Re[δ3↑ (t)] − Re[δ3↓ (t)]+ ),
2
(58)

=

↓

Fig. 4↑+ d e−Im[δ3 (t)]+Im[δ3 (t)] sin(Re[δ3↓ (t)] − Re[δ3↑ (t)] + π),(59)
d Im[δ3↓ (t)]
dt
+de

=

+Im[δ3↓ (t)]−Im[δ3↑ (t)]sin(Re[δ3↑ (t)] − Re[δ3↓ (t)]).
(60)

In the above differential equations, we set the value of dand the initial phase values at t = 0 as shown in Table 1.
Figure 3 plots the phases of Re[δ1↑ (t)], Re[δ1↓ (t)], Re[δ2↓ (t)],Re[δ2↑ (t)], Re[δ3↓ (t)] and Re[δ3↑ (t)]. The horizontal axis
represents time t. With temporal evolution, it is shown thatthe phase difference of Re[δ1↓ (t)] − Re[δ1↑ (t)] converges to
π/2 as shown in Fig. 2. However, the phase differencesof Re[δ2↓ (t)] − Re[δ1↑ (t)] and Re[δ2↑ (t)] − Re[δ1↑ (t)] do notconverge to π and 3π/2, respectively. The phase differenceof Re[δ3↓ (t)] − Re[δ3↑ (t)] converges to π/2. Although, Fig. 2
does not show the phase relation for Re[δ3↓ (t)], Re[δ3↑ (t)], wenote that Re[δ3↓ (t)] − Re[δ3↑ (t)] converges to π/2 from (57)and (58). In the next section, we will give a detailed analysis
about the convergence of the phase differences of modes 1,

Numerial results of Im[δ1↑ (t)], Im[δ1↓ (t)], Im[δ2↑ (t)], Im[δ2↓ (t)],Im[δ3↑ (t)] and Im[δ3↓ (t)].

2, and 3.
Figure 4 shows the numerical results of imaginaryparts Im[δ1↑ (t)], Im[δ1↓ (t)], Im[δ2↑ (t)], Im[δ2↓ (t)], Im[δ3↑ (t)]and Im[δ3↓ (t)]. The horizontal and vertical axis represent time and values of the imaginary parts, respectively.
With temporal evolution, it is shown that values of theimaginary parts decrease in proportion to time. According to Sect. 4, Im[δ1↑ (t)], Im[δ1↓ (t)], Im[δ2↑ (t)], Im[δ2↓ (t)],Im[δ3↑ (t)] and Im[δ3↓ (t)] decrease after synchronization ofRe[δ1↑ (t)], Re[δ1↓ (t)], Re[δ2↑ (t)], Re[δ2↓ (t)], Re[δ3↑ (t)] andRe[δ3↓ (t)]. These results show that the decreases in the values of the imaginary parts are not bound, and indeed continue
with time. By applying the numerical results of Im[δ1↑ (t)]↑↓and Im[δ1↓ (t)] to (51) and (52), e−Im[δ1 (t)] and e−Im[δ1 (t)] continue to increase exponentially with time. In the same way, it↑↓↑↓
is shown that e−Im[δ2 (t)] , e−Im[δ2 (t)] , e−Im[δ3 (t)] and e−Im[δ3 (t)]continue to increase exponentially with time.

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Table 2
ParameterdRe[δ1↑ (0)]Re[δ1↓ (0)]Re[δ2↑ (0)]Re[δ2↓ (0)]Re[δ3↑ (0)]Re[δ3↓ (0)]Fig. 5 Relationships of phase difference from each differential equation
after synchronization.

5.

Relationship between the Initial Phase and Divergence of the Degenerated Oscillation Modes

In this section, we show the relationship between the initial
phase and divergence of the degenerated oscillation modes.
5.1

Theoretical Analysis

Figure 5 shows the relationship of expected mode synchronization plots from the temporal evolution equations of oscillation modes 1, 2, and 3. In Figs. 5(a), (b), and (c), a circle
represents the unit circle in the complex plane. Those relations are obtained from the equations of phase differences ofoscillation modes 1, 2, and 3, respectively. As the absolute
position on the unit circle has no import, we focus on just thephase difference [16]. If the arguments of the sine functions
become 0 from (47) and (48) due to Kuramoto-like synchronization, the relationship of Fig. 5(a) is obtained. The
relationships in Fig. 5(b), and (c) are obtained in the samemanner.
Although Fig. 5(a) is obtained from Eqs. (47) and (48)for Re[δ1↑ (t)] and Re[δ1↓ (t)], it is also associated with thephase differences for Re[δ2↑ (t)] and Re[δ2↓ (t)]. In addition, (b) is obtained from the equations for Re[δ2↑ (t)] andRe[δ2↓ (t)], it is also associated with the phase differencesfor Re[δ3↑ (t)] and Re[δ3↓ (t)]. Note that (a), (b), and (c) in
Fig. 5 do not occur simultaneously due to the contradictory
phase relationship. For example, the phase of Re[δ2↑ (t)] is
advanced by π/2 from the phase of Re[δ2↓ (t)] in Fig. 5(a).
However, the phase of Re[δ2↑ (t)] is delayed by π/2 fromthe phase of Re[δ2↓ (t)]. Therefore, the phase relation ofRe[δ2↑ (t)] and Re[δ2↓ (t)], which satisfies Fig. 5(a) and (b) simultaneously, can not exist. Similarly, the phase relationof Re[δ3↑ (t)] and Re[δ3↓ (t)] which satisfies Fig. 5(b) and (c)simultaneously can not exist.
In order to prevent the divergence of the oscillationmode, it is sufficient to prevent the temporal evolution
equation from falling into Kuramoto-like synchronization.
Given oscillation mode 1, based on the synchronized statein Fig. 5(a) which is obtained from Eqs. (47) and (48) forRe[δ1↑ (t)] and Re[δ1↓ (t)], we can set the initial phase as thereverse phase, which is considered to be most difficult to synchronize. However, if we provide the reverse phase of theInitial phase parameters.
Value
1
−π/4
−3π/4
−3π/4
−π/4
−π/4
−3π/4

Parameter

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

0
0
0
0
0
0

synchronized state for oscillation mode 1, the phase for oscillation mode 2 is more likely to realize synchronization. On
the right-hand side of the second term of Eq. (47), Re[δ2↓ (t)]synchronizes with Re[δ1↑ (t)] by π. If the phase of Re[δ2↓ (t)]is advanced from the synchronized state by π to avoid thesynchronization in Fig. 5(a), oscillation mode 2 enters thesynchronized state because Re[δ2↑ (t)] is delayed by π/2 relative to Re[δ2↓ (t)] as shown in Fig. 5(b). This result indicatesthat preventing the synchronization of all phase differences
simultaneously is difficult for all combinations of the initial
phases.
If the real parts of phase difference trigger Kuramotolike synchronization, the values of the imaginary parts decrease in proportion to time, and the oscillation energy diverges. If any of the oscillation modes diverge, the oscillationenergy diverges. As a result, it can be seen that divergenceof the oscillation energy can be triggered by the degenerated
oscillation modes without regard to the initial phase.
5.2

Numerical Evaluation of Behavior of Oscillation
Modes

In this subsection, we evaluate the behavior of the oscillation
modes created by differences in the initial phases. By numerical simulation, we evaluate the behavior of the solutionof differential equations for the three oscillation modes.
In the first evaluation, we evaluate the behavior of thesolution on the condition that the initial states of the phasedifferences are set to the reverse phases of the synchronizedstates. However, as discussed in Sect. 5.1, we can not provide the reverse phases of the synchronized states for all
oscillation modes at the same time. Therefore, we set therelationship of the phase difference of mode 1 and 3 to thereverse phases of the synchronized state. On the other hand,
the relationship of the phase difference of mode 2 is set tothat of the synchronized state. Table 2 shows the initial phaseparameters used in the numerical simulations.
Figure 6(a) plots the real parts of phase of the oscillation modes over time. Oscillation mode 1 (Re[δ1↑ (t)],Re[δ1↓ (t)]) triggered phase synchronization. Oscillation
mode 2 (Re[δ2↑ (t)], Re[δ2↓ (t)]) did not trigger phase synchronization because the initial state already realized phasesynchronization. Oscillation mode 3 (Re[δ3↑ (t)], Re[δ3↓ (t)])
also did not trigger phase synchronization and maintained
the initial state continuously.

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Fig. 7

Phase difference of mode 1 after sufficient time (a)Re[δ1↓ (t)] −Re[δ1↑ (t)], (b)Re[δ2↓ (t)] − Re[δ1↑ (t)], (c)Re[δ2↑ (t)] − Re[δ1↓ (t)].

Fig. 8Phase difference of mode 2 after sufficient time (a)Re[δ2↓ (t)] −Re[δ2↑ (t)], (b)Re[δ3↓ (t)] − Re[δ2↑ (t)], (c)Re[δ3↑ (t)] − Re[δ2↓ (t)].
Fig. 6 Numerical results of real and imaginary parts of the phases ofoscillation modes.

Figure 6(b) shows the imaginary parts of the phasesof the oscillation modes. with attention to Im[δ3↓ (t)]; it
increases in proportion to time. This result indicates thatthe sine function of mode 3 did not become minus because
Re[δ3↑ (t)] and Re[δ3↓ (t)] maintained the initial phase. However, the sine function of mode 2 held −1 because Re[δ2↑ (t)]and Re[δ2↓ (t)] maintained the initial phase. Therefore, withtemporal evolution, it is shown that the imaginary parts ofoscillation modes 1 and 2, which Re[δ2↑ (t)] and Re[δ2↓ (t)]contribute to, decrease in proportion to time. From this result, we can confirm divergence of the solutions over timeeven if the initial phases were the reverse of those in thesynchronized state.
We evaluated the behavior of more diverged oscillation
modes. We performed 1 × 104 times trials in which thereal parts of the initial phases of oscillation modes 1, 2,
and 3 at t = 0 were randomly set to a number between
0 and 2π. Figure 7 shows a histogram of the real partsof the phase differences Re[δ1↓ (t)] − Re[δ1↑ (t)], Re[δ2↓ (t)] −Re[δ1↑ (t)], and Re[δ2↑ (t)]−Re[δ1↓ (t)], that are related to mode
1, after sufficient time. In Fig. 7(a), the phase differenceRe[δ1↓ (t)] − Re[δ1↑ (t)] is mostly distributed around π/2, andexhibits Kuramoto-like synchronization.
On the other hand, Fig. 7(b) shows that the phase difference Re[δ2↓ (t)] − Re[δ1↑ (t)] is widely distributed althoughsome bias is observed for π. Figure 7(c) shows that thephase difference Re[δ2↑ (t)] − Re[δ1↓ (t)] is widely distributed
although some bias is observed for π, see Fig. 7(b). In thesecases, signs of the sine function of the imaginary part are
not specified. Therefore, it is unclear whether Im[δ1↑ (t)] orIm[δ1↓ (t)] decrease in proportion to time. Figure 8 shows ahistogram of the real parts of the phase difference related tomode 2. Figure 8(b), (c) follow Fig. 7(b), (c). The differenceis that Fig. 8(a) shows wider distributions than Fig. 7(a).
Although one of three oscillation modes does not contribute to the divergence of the oscillation energy, our interest is the possibility of oscillation energy divergence as awhole if the phase synchronization of other oscillation mode

KUBO et al.: NEW MODEL OF FLAMING PHENOMENA IN ON-LINE SOCIAL NETWORKS CAUSED BY DEGENERATED OSCILLATION MODES

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Fig. 9Phase difference of mode 3 after sufficient time Re[δ3↓ (t)] −Re[δ3↑ (t)].

contributes to the divergence of the oscillation energy. To
examine the above possibility, we show the phase differenceRe[δ3↓ (t)]−Re[δ3↑ (t)], which is related to the sine function of
the oscillation mode 3, in Fig. 9. From Fig. 9, the sine function turns negative because Re[δ3↓ (t)] − Re[δ3↑ (t)] + π, whichis the parenthesis of the sine function in (59), synchronizes
to 3π/2. In a similar way, the sine function turned negative
because Re[δ3↑ (t)] − Re[δ3↓ (t)], which is the parenthesis of
the sine function in (59), synchronizes to 3π/2. As a result,the sine function in the right hand side of (59) and (60) is −1.
Then, the oscillation energy of mode 3 diverges because theimaginary part of the phase decreases. These results showthat the degenerated oscillation modes can force divergenceof the oscillation energy regardless of the initial phase.
6.

Conclusion

This paper showed that the flaming phenomena can occur
due to coupling between degenerated oscillation modes even
if all eigenvalues are real numbers; this finding is quite different from previous studies on the flaming phenomenon
which assume that the eigenvalues must be imaginary numbers. We adopted the Jordan canonical form of the linear
operator corresponding to network structure to analyze thesolution. This yields a simplified expression as the fundamental equation of the oscillation model on networks. If thelinear operator cannot be diagonalized, we confirmed that theflaming phenomena can still occur due to oscillation mode
divergence after the real parts of the phases of the degenerated oscillation modes synchroniz. Numerical simulations
clearly showed the divergence of the solutions with time. We
also investigated the relationship between the initial phasesof the degenerated oscillation modes and oscillation energy
divergence. Numerical simulations showed that divergenceof the oscillation energy is independent of the initial phasesof the degenerated oscillation modes.
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Takahiro Kubo
received a B.S. degreein Physics from Tokyo Metropolitan University,
Japan, in 2005, and a M.S. in Multidisciplinary
Sciences from the University of Tokyo, Japan,
in 2007. In 2007, he joined NTT Access Network Service Systems Laboratories, where heengaged in research on optical access systems
using free space optics technologies. From 2013
to 2016, he was with NTT DOCOMO, INC.,
where he worked on mobile radio access technologies. He is currently a graduate student,
studying toward his Ph.D. degree at the Graduate School of Systems Design, Tokyo Metropolitan University. His current interest includes analysis
of social network dynamics. He received the IEICE Young Researchers’Award in 2015. He is a member of the IEEE and IEICE.

IEICE TRANS. COMMUN., VOL.E102–B, NO.8 AUGUST 2019

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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. Since April
2008, she has been an Associate Professor at theGraduate School of Information Sciences, Hiroshima City University. Her research interests arecomputer networks and distributed systems. She
received the IEICE Young Researchers’ Award in 2003. She received theInformation Network Research Awards from the IEICE in 2004, 2012, and2015. She is a member of IEEE (U.S.A.).

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 University ofTokyo, Japan, in 1999. In April 1989, he joined
NTT Laboratories. From April 2005 to March
2007, he was an Associate Professor at the Faculty of System 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 computer communication
networks. He received the Best Tutorial Paper Award and the Best Paper
Award of IEICE Communications Society in 2013 and 2016, respectively.
He also received the IEICE 100-Year Memorial Paper Award in 2017. He
is a member of the IEEE, ACM and the Operations Research Society ofJapan.