2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
1

Technology to Counter Online Flaming Based on the
Frequency-Dependent Damping Coeﬃcient in the Oscillation Model
Shinichi KIKUCHI†a) , Student Member, Chisa TAKANO††b) , Member, and Masaki AIDA†c) , Fellow

SUMMARY Online social networks, which are remarkably active, often experience explosive user dynamics such as online ﬂaming, which cansigniﬁcantly impact the real world. However, countermeasures based on social analyses of the individuals causing ﬂaming are too slow to be eﬀective
because of the rapidity with which the inﬂuence of online user dynamics propagates. A countermeasure technology for the ﬂaming phenomena
based on the oscillation model, which describes online user dynamics, hasbeen proposed; it is an immediate solution as it does not depend on social
analyses of individuals. Conventional countermeasures based on the oscillation model assume that the damping coeﬃcient is a constant regardlessof the eigenfrequency. This assumption is, however, problematic as thedamping coeﬃcients are, in general, inherently frequency-dependent; thetheory underlying the dependence is being elucidated. This paper discusses
a design method that uses the damping coeﬃcient to prevent ﬂaming undergeneral conditions considering the frequency-dependence of the damping
coeﬃcient and proposes a countermeasure technology for the ﬂaming phenomena.
key words: online ﬂaming, user dynamics, oscillation model, damping
coeﬃcient

1. Introduction
In recent years, with the spread of social networking sites
such as Twitter and Facebook, users’ activities in online social networks have come to be closely connected to social
activities in the real world, not only in online communities.
As a result, the eﬀects of explosive online user dynamics, including the ﬂaming phenomena, are becoming more serious,and countermeasures are needed [1], [2].
Although it is desirable to respond immediately withdirect countermeasures to eliminate the factors that cause
the ﬂaming phenomena, analyzing each event in detail, one
by one, will be too slow to prevent the damage from spreading [3], [4]. Thus we need an engineering framework for
ﬂaming countermeasures that does not depend on the details of each individual event. One such framework has
been proposed [5]. This is based on the oscillation modelon networks [6], [7] which is used to describe online user
dynamics.
Conventional countermeasures for the ﬂaming phenomena have been discussed under the assumption that the damping coeﬃcient [5] is a constant, and independent of the eigenfrequency. However, it is known that regardless of the phe† The authors are with Tokyo Metropolitan University, Hino,
Tokyo, 191-0065 Japan.
†† The author is with Hiroshima City University, Hiroshima,731-3194 Japan.
a) E-mail: kikuchi-shinichi@ed.tmu.ac.jpb) E-mail: takano@hiroshinma-cu.ac.jpc) E-mail: aida@tmu.ac.jp

nomenon, the damping coeﬃcient generally depends on the
eigenfrequency. In fact, the theoretical characteristics of thefrequency dependence of the damping coeﬃcient have recently been clariﬁed [8]. Based on these insights, we canconsider countermeasures for the ﬂaming phenomena basedon the oscillation model, even in general cases where thedamping coeﬃcient does depend on the eigenfrequency.
In this paper, we introduce a design methodology thatallows the damping coeﬃcient to be used to counter the ﬂaming phenomena even when the damping coeﬃcient depends
on the eigenfrequency; we use it to propose a countermeasuretechnology for the ﬂaming phenomena.
2.

Oscillation Model for Describing Online User Dynamics

Let the Laplacian matrix of the online social network (OSN)with 𝑛 nodes be L, which is an 𝑛 × 𝑛 square matrix, and theweight of the link from node 𝑖 to node 𝑗 (𝑖 → 𝑗) be 𝑤𝑖 𝑗 . In
addition, let the eigenvalues of L be 𝜆 𝜇 (𝜇 = 0, 1, . . . , 𝑛−1)
and the eigenvectors associated with 𝜆 𝜇 be 𝒗 𝜇 . We assume
the eigenvalues are not duplicated.
The eigenvalues of L are generally complex numbers,
whose range of existence is given by the largest Gershgorin
disk[9] of L as
𝐷 max
𝐺 (L) = {𝑧 ∈ C : |𝑧 − 𝑑 max | ≤ 𝑑 max } ,(1)where 𝑑max is the maximum weighted out-degree of the network. It is known that all the eigenvalues of L lie within its
largest Gershgorin disk [5].
The oscillation model [5], [6] is a minimal model for
describing user dynamics in OSNs. Let 𝑥𝑖 (𝑡) be the stateof node (user) 𝑖 at time 𝑡. Since the inﬂuence of interaction between users must propagate through any OSN at
a ﬁnite speed, its description by the wave equation should
be possible, which is the equation for describing the propagation of something at ﬁnite speed. For the state vector
𝒙(𝑡) := 𝑡(𝑥 1 (𝑡), . . . , 𝑥 𝑛 (𝑡)), the wave equation in the OSN
is written asd2d𝒙(𝑡) + 𝚪 𝒙(𝑡) = −L 𝒙(𝑡),
(2)
2
d𝑡
d𝑡
where 𝚪 is the matrix expressing the strength of the damping.
Substituting the expansion of 𝒙(𝑡) by 𝒗 𝜇 as𝒙(𝑡) =𝑛−1Õ

𝑎 𝜇 (𝑡) 𝒗 𝜇 ,𝜇=0Copyright © 2020 The Institute of Electronics, Information and Communication Engineers(3)

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)2

into the wave equation (4), yields the equation of motion foreach oscillation mode 𝑎 𝜇 (𝑡) (𝜇 = 0, 1, . . . , 𝑛 − 1) asd2
d𝑎 𝜇 (𝑡) + 𝛾(𝜔 𝜇 ) 𝑎 𝜇 (𝑡) = −𝜆 𝜇 𝑎 𝜇 (𝑡),
(4)
2
d𝑡
d𝑡
where 𝛾(𝜔 𝜇 ) is the damping coeﬃcient; it depends on 𝜔 𝜇 =p
𝜆 𝜇 and is expressed as𝛾(𝜔 𝜇 ) := 𝛾0 + 𝛾1 𝜆 𝜇

the absolute value of 𝑎 𝜇 (𝑡), we derive the condition underwhich 𝑎 𝜇 (𝑡) does not diverge and then the condition underwhich ﬂaming does not occur.
By decomposing the damping coeﬃcient 𝛾(𝜔 𝜇 ) into
real and imaginary parts as in𝛾(𝜔 𝜇 ) = 𝛾0 + 𝛾1 Re[𝜆 𝜇 ] + i 𝛾1 Im[𝜆 𝜇 ].
(9)𝑎 𝜇 (𝑡) is written as  
𝛾0 + 𝛾1 Re[𝜆 𝜇 ]𝜃𝜇√𝑎 𝜇 (𝑡) = 𝑐+𝜇 exp −𝑡 − 𝑟 𝜇 sin
𝑡
2
2

  
𝛾1 Im[𝜆 𝜇 ]𝜃𝜇√
× 𝑐+𝜇 exp −i
𝑡 + i 𝑟 𝜇 cos
𝑡
2
2

  
𝛾0 + 𝛾1 Re[𝜆 𝜇 ]𝜃𝜇√+ 𝑐−𝜇 exp −𝑡 + 𝑟 𝜇 sin
𝑡
2
2

  
𝛾1 Im[𝜆 𝜇 ]𝜃𝜇√
× 𝑐−𝜇 exp −i
𝑡 − i 𝑟 𝜇 cos
𝑡 .
2
2
(10)

with the constant 𝛾0 and 𝛾1 [8]. Note that Re[𝛾(𝜔 𝜇 )] =𝛾0 + 𝛾1 Re[𝜆 𝜇 ] ≥ 0.
The solution of (4) is written as 𝛾(𝜔 𝜇 )𝜃𝜇√𝑎 𝜇 (𝑡) = 𝑐+𝜇 exp −𝑡 + i 𝑟 𝜇 exp i
𝑡
2
2


 
𝛾(𝜔 𝜇 )
𝜃𝜇√
+ 𝑐−𝜇 exp −𝑡 − i 𝑟 𝜇 exp i𝑡 , (5)22where 𝑐+𝜇 and 𝑐−𝜇 are constants that depend on 𝜇, and 𝑟 𝜇 and𝜃 𝜇 (−𝜋 < 𝜃 𝜇 ≤ 𝜋) are, respectively, the absolute value andthe argument of the following complex number:𝛾(𝜔 𝜇 ) 2𝛾(𝜔 𝜇 ) 2=− 𝛼+.
𝑟 𝜇 exp(i 𝜃 𝜇 ) := 𝜆 𝜇 −22(6)In the oscillation model, the oscillation energy can beconsidered as the strength of the activity of user dynamics[10], [11]. Also, the situation in which oscillation energy
diverges over time is considered to describe explosive user
dynamics, which include the ﬂaming phenomena. Therefore,
in order to prevent explosive user dynamics, it is necessary
to consider the conditions under which the oscillation energy
does not diverge. By deriving the strength of the dampingthat satisﬁes this condition, we can obtain a framework inwhich the strength of damping can be adjusted to prevents
the ﬂaming phenomena.
The conventional solutions to the ﬂaming phenomena
assume that the damping coeﬃcient is a constant and independent of eigenfrequency. This corresponds to the special
case of 𝛾1 = 0 in (4). Following [5], the condition underwhich the oscillation energy does not diverge is given as
∀

𝜇,

𝜃𝜇
𝛾0
,
√ ≥ sin
2 𝑟𝜇
2

(7)

and the value of the damping coeﬃcient required to satisfy
this condition is given byp
𝛾0 ≥ 2 𝑑max .
(8)
In the next section, in order to consider countermeasures
for the ﬂaming phenomena, we discuss the conditions underwhich the oscillation energy does not diverge in the case of𝛾1 ≠ 0.

To determine if 𝑎 𝜇 (𝑡) diverges or not, we need to check
whether the real components of the exponent of the exponential function in (10) are positive or negative, and if 𝑎 𝜇 (𝑡)diverges, the following condition is satisﬁed:   Re[𝛾(𝜔 𝜇 )]𝜃𝜇Re[𝛾(𝜔 𝜇 )]
𝜃𝜇
+
sin
−
sin
< 0.
√
√
2 𝑟𝜇
2
2 𝑟𝜇
2
This means the one of real components of the exponent is
positive and the other is negative.
Consequently, the condition under which the oscillationenergy diverges is given by∃𝜇,𝛾0 + 𝛾1 Re[𝜆 𝜇 ]
𝜃𝜇
< sin
,
√
2 𝑟𝜇
2

(11)

and the condition that the oscillation energy does not diverge
is obtained as∀

𝜇,𝛾0 + 𝛾1 Re[𝜆 𝜇 ]
𝜃𝜇
≥ sin
.
√
2 𝑟𝜇
2

(12)

Since the conventional condition (7) that the oscillation energy does not diverge corresponds to the case of 𝛾1 = 0
as per condition (12), the condition (12) for the frequencydependent damping coeﬃcient is a generalization of the conventional result.
4.

Countermeasure for Flaming Phenomena Given the
Frequency-Dependent Damping Coeﬃcient

Based on condition (12), i.e., the oscillation energy does
not diverge, we consider a design method for the damping
coeﬃcient to satisfy (12), and consider a countermeasuretechnology for the ﬂaming phenomena by adjusting the valueof the damping coeﬃcient.

3. Model of Explosive User Dynamics Considering
Frequency-Dependent Damping Coeﬃcient

4.1

Since the oscillation energy is proportional to the square ofAmong parameters 𝛾0 and 𝛾1 , which determine the strength

Adjusting the Damping Coeﬃcient2020 International Conference on Emerging Technologies for Communications (ICETC 2020)3of damping, 𝛾1 is the eigenfrequency dependent term. Thismeans that the value of 𝛾1 is a parameter predetermined by
the structure of the network. Therefore, 𝛾0 is the only parameter that can be manipulated independently of the network
structure. In this framework, even if various values are given
as 𝛾1 , we can counter the ﬂaming phenomena by adjusting
the value of 𝛾0 . Here, the actual action to adjust the value of
𝛾0 includes disseminating other information to attract users’attention. In the following, we consider the value of 𝛾0 necessary to prevent the triggering of explosive user dynamics,
and we use its value in a ﬂaming countermeasure.
The range of eigenvalues of L is the interior of thelargest Gershgorin disk (including its boundaries) of radius𝑑max with center (𝑑max , 0). From condition (12), the oscillation energy does not diverge, we can consider the range
satisfying condition (12) on the complex plane. Then, if thelargest Gershgorin disk of L lies completely within the range
on the complex plane, the oscillation energy never diverges
regardless of the network structure.
In order to clarify the region on the complex plane inwhich condition (12) ensures that the oscillation energy does
not diverge, the inequality of condition (12) is transformed
as follows:Im[𝜆 𝜇 ] 2 ≤𝑍,
4 − 4𝛾0 𝛾1 − 4 𝛾12 Re[𝜆 𝜇 ]𝛾02 + 2 𝛾0 𝛾1 𝑑max − 2 𝑑max ≥ 0.

(16)

The range of 𝛾0 that satisﬁes the above is as follows
from 𝛾0 ≥ 0q2
𝛾0 ≥ 𝛾12 𝑑max
+ 2 𝑑max − 𝛾1 𝑑max .
(17)• if 𝛾0 𝛾1 + 1 + 2𝛾12 𝑑max > 0,
𝛾02 + 2 𝛾0 𝛾1 𝑑max − 2 𝑑max
𝛾0 𝛾1 + 1 + 2 𝛾12 𝑑max

≥ −Re[𝜆 𝜇 ].

(18)

In order for this inequality to hold, the numerator needs
to be non-negative, so we obtain
𝛾02 + 2 𝛾0 𝛾1 𝑑max − 2 𝑑max ≥ 0.

(19)

Considering 𝛾0 ≥ 0, the condition of 𝛾0 to ensure thenon-divergence of oscillation energy for all eigenvalues
is written asq2+ 2 𝑑max − 𝛾1 𝑑max .
(20)
𝛾0 ≥ 𝛾12 𝑑max
• if 𝛾0 𝛾1 + 1 + 2 𝛾12 𝑑max < 0,𝛾02 + 2 𝛾0 𝛾1 𝑑max − 2 𝑑max

(13)

𝛾0 𝛾1 + 1 + 2 𝛾12 𝑑max

where

≤ −Re[𝜆 𝜇 ].

(21)

Inequality (21) is transformed to𝑍 = 𝛾04 + 4𝛾03 𝛾1 Re[𝜆 𝜇 ] + 6 𝛾02 𝛾12 Re[𝜆 𝜇 ] 2+ 4 𝛾02 𝑋 + 4 𝛾0 𝛾13 Re[𝜆 𝜇 ] 3 + 8 𝛾0 𝛾1 Re[𝜆 𝜇 ] 𝑋+ 𝛾14 Re[𝜆 𝜇 ] 4 + 4 𝛾12 Re[𝜆 𝜇 ] 2 𝑋,

(𝛾0 + 2 𝛾1 𝑑max ) 2 ≥ 0

Therefore, inequality (21) always holds in this case.

and 𝑋 is the real part of (6).
The range of eigenvalues of L, determined by the Gershgorin theorem, are written as
2Im[𝜆 𝜇 ] 2 ≤ 𝑑max
− (Re[𝜆 𝜇 ] − 𝑑max ) 2 .

(14)

We compare (14) and the condition (13) that the oscillationenergy does not diverge. Because both of them are axiallysymmetric on by the real axis, we consider the upper-halfplane. The condition that the largest Gershgorin disk iscompletely included the range of (13) is expressed as
2𝑑max− (Re[𝜆 𝜇 ] − 𝑑max ) 2 ≤

𝑍,
4 (1 − 𝛾0 𝛾1 − 𝛾12 Re[𝜆 𝜇 ])which can be transformed to
𝛾0 𝛾1 + 1 + 2 𝛾12 𝑑max Re[𝜆 𝜇 ]≥ −(𝛾02 + 2 𝛾0 𝛾1 𝑑max − 2 𝑑max ),(15)

by considering Re[𝜆 𝜇 ] ≥ 0.
We consider the conditions for satisfying inequality (15)in the following three cases.
• if 𝛾0 𝛾1 + 1 + 2𝛾12 𝑑max = 0,(22)

To summarize the above results, the 𝛾0 condition that
ensures the oscillation energy does not diverge for all eigenvalues is obtained asq2
𝛾0 ≥ 𝛾12 𝑑max
+ 2 𝑑max − 𝛾1 𝑑max .
(23)
Therefore, given the maximum weighted out-degree ofthe network, 𝑑max , and parameter 𝛾1 of the damping coeﬃcient, adjusting the value of 𝛾0 to satisfy (23) will counterthe ﬂaming phenomena.
4.2

Case Studies

Using an example network with 𝑑max = 100, this section
considers three cases of diﬀerent values of 𝛾1 , the frequency
dependence of the damping coeﬃcient: 𝛾1 = 0, 𝛾1 > 0or 𝛾1 < 0. In all cases, we conﬁrm that the region of
the condition that the oscillation energy does not diverge
includes the largest Gershgorin disk of L by satisfying thecondition (23) of 𝛾0 .
First, we conﬁrm
the case of complete ﬂaming prevenq

2tion with 𝛾0 = 𝛾12 𝑑max
+ 2 𝑑max − 𝛾1 𝑑max , where 𝛾0 is theminimum value that satisﬁes condition (23). Figure 1 shows
the regions in which the oscillation energy does not diverge
as given by condition (12), for the cases of 𝛾1 = 0.1, 𝛾1 = 0

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
4

Fig. 1

Examples of complete ﬂaming prevention

Fig. 2

Examples of incomplete ﬂaming preventionand 𝛾1 = −0.1. These regions are depicted in blue. In addition, the largest Gershgorin disk of L is depicted in red. In
all ﬁgures, it can be seen that the regions wherein the oscillation energy does not diverge completely include the largest
Gershgorin disk, so that no divergence of oscillation energy
occurs regardless of the details of the network structure.
Next, we showqthe case of incomplete ﬂaming preven-2tion by using 𝛾0 = 𝛾12 𝑑max
+ 2 𝑑max − 𝛾1 𝑑max − 5, in which
𝛾0 is less than the minimum value that satisﬁes condition(23). Figure 2 shows the regions wherein the oscillation energy does not diverge as indicated by condition (12), for thecases of 𝛾1 = 0.1, 𝛾1 = 0, and 𝛾1 = −0.1. In these cases,
the regions cannot completely enclose the Gershgorin disk.
If even just one eigenvalue appears outside of the region, theoscillation energy diverges and the ﬂaming phenomenon occurs. Therefore, depending on the position of the eigenvaluesof L, ﬂaming prevention is not assured.

5. Conclusion
In this paper, we proposed a countermeasure technology forthe ﬂaming phenomena based on the oscillation model withthe frequency-dependent damping coeﬃcient. The design
method that yields the damping coeﬃcients using condition(23) is a generalized version of the conventional countermeasure technology for the ﬂaming phenomena. Regardlessof the value of parameter 𝛾1 , which is the strength of thefrequency dependence of damping coeﬃcient, we can prevent explosive user dynamics by setting parameter 𝛾0 to bean appropriate value.
Furthermore, the required value of 𝛾0 can be determinedfrom just 𝑑max , which is the maximum weighted out-degreeof the network, and 𝛾1 , which is the strength of the frequencydependence of the damping coeﬃcient. One of the methods
for increasing the value of 𝛾0 in the actual OSNs is that todisseminate other information to attract users’ attention is

mentioned.
Acknowledgement
This research was supported by Grant-in-Aid for ScientiﬁcResearch 19H04096 and 20H04179 from the Japan Society
for the Promotion of Science (JSPS).
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