2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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Independence of the Fundamental Equation of the Oscillation
Model on Algebraic Representations: Social Media Echo Chamber
Eﬀect
Kakeru OHKI†a) , Ayako HASHIZUME††b) , Nonmembers, and Masaki AIDA†c) , Fellow

SUMMARY In the oscillation model that describes the user dynamics
of online social networks, it is known that the fundamental equation can
explicitly describe the causal relationship between the network structure anduser dynamics. The fundamental equation uses algebra that satisﬁes the anticommutation relation, and its matrix representation is not unique. However,even if the matrix representations are diﬀerent, the same results should bederived from diﬀerent representations of the fundamental equation if they
are describing the same phenomenon. In this paper, we conﬁrm, using
the echo-chamber eﬀect as an example, that the fundamental equations ofdiﬀerent matrix representations lead to the same result.
key words:
oscillation model, social media echo chamber, anticommutation relation, fundamental equation

1. Introduction
With the rapid development of information networks inrecent years, social networking services (SNSs) have become widespread, and user interest in online social networks(OSNs) has become active. While such user dynamics in
OSN support human activities in the real world by promoting information exchange and mutual understanding, theyalso cause social problems such as the online ﬂaming phenomenon and echo-chamber eﬀect. Therefore, understanding user dynamics in OSN is a crucial issue.
The oscillation model is known to be able to describe
user dynamics on OSNs [1], [2]. This is a minimal model thatapplies the wave equation on networks. The wave equation
is an equation that expresses the phenomenon that an object
propagates through a medium at ﬁnite speed. In the conceptof the oscillation model, the strength of the users’ activity
should have some inﬂuence on each other through the OSN,
so the wave equation models the propagation of the inﬂuence
through the OSN at ﬁnite speed.
The concept of node centrality represents the strength
and/or importance of the activity of nodes in the network[3]–[5]. The characteristic of the oscillation model is thatthe oscillation energy gives a generalized concept of node
centrality and so can express the strength of network activity[6], [7]. Conventionally, typical node centralities used in network analysis are the degree centrality and the betweenness† The authors are with Tokyo Metropolitan University, Hino,
Tokyo, 191-0065 Japan.
†† The author is with Hosei University, Machida, Tokyo, 1940298 Japan.
a) E-mail: ohki-kakeru@ed.tmu.ac.jpb) E-mail: hashiaya@hosei.ac.jpc) E-mail: aida@tmu.ac.jp

centrality; fortunately, the oscillation model can provide a
uniﬁed framework for both of them. That is, under the condition that there is no bias in network usage, all link weights
are assumed to be 1, the oscillation energy of each node
yields degree centrality. On the other hand, taking the linkweight to be the number of shortest paths that pass throughthat link (e.g., the volume of traﬃc), the oscillation energyof each node leads to the betweenness centrality. Furthermore, the oscillation model can be applied even when the
network usage is biased, and provides a generalization ofnode centrality.
The oscillation model makes it possible to derive afundamental equation that can explicitly describe the causal
relationship between the structure of the network and theuser dynamics generated from the structure. However, since
the method of representing the fundamental equations using
a matrix is not unique, it is necessary to conﬁrm whether thefundamental equations in diﬀerent matrix representations
describing the same phenomenon yield the same results. If
we get the same result, the fundamental equation can beconsidered to be well-deﬁned. This paper takes the echochamber eﬀect as an example and shows that the same user
dynamics can be derived from the fundamental equations in
diﬀerent matrix representations.
The rest of this paper is organized as follows. Sec. 2
gives an overview of the oscillation model and explains thefundamental equation. In Sec. 3, we show the characteristics
of user dynamics representing the echo-chamber eﬀect thatis derived by using the fundamental equation. In Sec. 4,
another representation of the fundamental equation is introduced, the user dynamics representing the echo-chamber
eﬀect is derived by using the other representation. The results show that the same user dynamics as described in Sec. 3
are obtained. Finally, Sec. 5 concludes this paper.
2.

Oscillation Model for User Dynamics in OSNs

The fundamental equations of user dynamics can be summarized as follows. First, for nodes 𝑖, 𝑗 ∈ 𝑉 of directed
graph 𝐺 (𝑉, 𝐸) that represents the structure of an OSN with𝑛 users, if the weight of directed link (𝑖 → 𝑗) ∈ 𝐸 is given
as 𝑤𝑖 𝑗 , the adjacent matrix A = [A𝑖 𝑗 ] 1≤𝑖, 𝑗 ≤𝑛 is deﬁned as{
𝑤𝑖 𝑗 ,(𝑖 → 𝑗) ∈ 𝐸,
A𝑖 𝑗 :=(1)
0,(𝑖 → 𝑗) ∉ 𝐸 .

Copyright © 2020 The Institute of Electronics, Information and Communication Engineers

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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Also, given nodal (weighted) out-degree 𝑑𝑖 :=the degree matrix is deﬁned as
D := diag(𝑑1 , . . . 𝑑 𝑛 ).

∑𝑗 ∈𝜕𝑖 𝑤𝑖 𝑗 ,(2)Here, 𝜕𝑖 denotes the set of adjacent nodes of out-links fromnode 𝑖. Next, the Laplacian matrix of the directed graphrepresenting the structure of the OSN is deﬁned by
L := D − A.

(3)

Let the state vector of users at time 𝑡 be𝒙(𝑡) := 𝑡(𝑥1 (𝑡), . . . , 𝑥 𝑛 (𝑡)),where 𝑥𝑖 (𝑡) (𝑖 = 1, . . . , 𝑛) denotes the user state of node 𝑖
at time 𝑡. Then, the wave equation for the OSN is written asd2𝒙(𝑡) = −L 𝒙(𝑡).
d𝑡 2

(4)

Here, in addition to simply ﬁnding the solution 𝒙(𝑡) of the
wave equation (4), it is desirable to be able to describe whatkind of OSN structure impacts user dynamics. In otherwords, we want to describe the causal relationship between
OSN structure and user dynamics. To achieve this, we needto develop a ﬁrst-order diﬀerential equation with respect
to time (hereinafter referred to as the fundamental equation) [8], [9].
Let us introduce a new matrix, H, as follows.
√√√
H := D−1 L = D − D−1 A,
(5)
√
√
√
where D := diag( 𝑑1 , . . . , 𝑑 𝑛 ). As is well-known, thenormalized Laplacian matrix is deﬁned as√√√√
N := D−1 L D−1 = 𝑰 − D−1 A D−1 ;so we call H the semi-normalized Laplacian matrix. Here,𝑰 is the 𝑛 × 𝑛 unit matrix.
Using the semi-normalized Laplacian matrix H, a new
fundamental equation for user dynamics can be written as
follows.
()√
d
i 𝒙(𝑡)
ˆ = H ⊗ 𝒂ˆ + D ⊗ 𝒃ˆ 𝒙(𝑡),
ˆ
(6)
d𝑡

𝒂ˆ 2 = 𝒃ˆ 2 = ô,(8)where 𝒆ˆ denotes the 2 × 2 unit matrix and ô denotes the nullmatrix.
3. Echo-Chamber Eﬀect
The echo-chamber eﬀect is a phenomenon in which beliefs
that are far from common sense are strengthened withina relatively small community existing in an online social
network. One model proposed to explain the occurrence ofthe echo-chamber eﬀect posits that a relatively small partial
network corresponding to a closed community detaches itself
from the online social network, and the separated subnetwork
becomes a complete graph [12], [13].
Here, based on the fundamental equation (6), let us
consider the situation in which the weights of all the links
in the separated subnetwork have the same value of 𝑤 [14].
This is a situation in which the weight of the link, whichindicates the strength of the relationship between users, has
increased to the limit and is saturated because the discussionis activated in the divided community. At this time, if thenumber of users in the community is 𝑛, the nodal degree
is the same for all nodes, and 𝑑 = (𝑛 − 1) 𝑤. Also, inthe corresponding Laplacian matrix, all eigenvalues otherthan 0 are duplicated, and the eigenvalues are denoted as
𝜆 = 𝜔2 = 𝑛 𝑤.
In this situation, since the degree matrix is D = 𝑑 𝑰,
which is√proportional to the identity matrix 𝑰, the matrices
H and D appearing in the fundamental equation (6) can
exhibit simultaneous diagonalization. Therefore, if the fundamental√ equation (6) is transformed so that the matrices H
and D are diagonalized simultaneously and expressed in
the block diagonalized form of 2 × 2, the equations for all the
eigenvalues, other than the eigenvalue of 0 of the Laplacianmatrix, are the same and can be written as follows:
(
])
[
] √ [
𝜔2 +1 +1
𝑑 +1 −1di 𝝍(𝑡) =
+
𝝍(𝑡)
√
d𝑡
2 +1 −1
2 𝑑 −1 −1[ 𝜔2 +𝑑]2+ √+ 𝜔 √−𝑑
2 𝑑
2 𝑑
=
𝝍(𝑡).
(9)
2
2
− 𝜔 √−𝑑 − 𝜔 √+𝑑
2 𝑑

2 𝑑

Here, if the two-dimensional vector of the solution of (9) isdenoted as
( + )
𝜓 (𝑡)
𝝍(𝑡) = −
,
(10)
𝜓 (𝑡)

where 𝒂ˆ and 𝒃ˆ are 2 × 2 matrices deﬁned as[][]
1 +1 −1
1 +1 +1
ˆ
, 𝒃=
,
𝒂ˆ =
2 −1 −1
2 +1 −1and 𝒙(𝑡)ˆis the 2𝑛-dimensional state vector. The solution
𝒙(𝑡) of the original wave equation (4) can be obtained fromthe solution 𝒙(𝑡)
ˆ by𝒙(𝑡) = ( 𝑰 ⊗ (1, 1)) 𝒙(𝒕).
ˆˆ := 𝒂ˆ 𝒃ˆ + 𝒃ˆ 𝒂ˆ = 𝒆,
{ 𝒂,
ˆ 𝒃}
ˆ

(7)

⊗ denotes the Kronecker product [10].
The fundamental equation is similar to the Dirac equation of relativistic quantum mechanics, and its feature is that𝒂ˆ and 𝒃ˆ satisfy the anti-commutation relation:and set the Ansatz of (9) as()
𝜓 ± (𝑡) := exp ∓i𝜃 ± (𝑡) ,(11)

in double sign correspondence; this yields
( (
))
d ±
𝜔2 + 𝑑 𝜔2 − 𝑑𝜃 (𝑡) = √ + √ exp ±i 𝜃 + (𝑡) + 𝜃 − (𝑡) .
d𝑡
2 𝑑
2 𝑑
(12)

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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Since 𝜃 ± (𝑡) is a complex number in general, by substituting
𝜃 ± (𝑡) = Re[𝜃 ± (𝑡)] + i Im[𝜃 ± (𝑡)]into (12), we obtain the temporal evolutions of the real andthe imaginary parts of 𝜃 ± (𝑡) asdRe[𝜃 ± (𝑡)]d𝑡()
𝜔2 + 𝑑= √ + 𝐶 ± (𝑡) cos Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)]2 𝑑
𝜔2 + 𝑑
= √
2 𝑑
( (
)
𝜋)
+ 𝐶 ± (𝑡) sin − Re[𝜃 ∓ (𝑡)] −− Re[𝜃 ± (𝑡)] ,
2
(13)
(
)
d
Im[𝜃 ± (𝑡)] = ±𝐶 ± (𝑡) sin Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)] ,
d𝑡
(14)
where
𝐶 ± (𝑡) :=( ())
𝜔2 − 𝑑exp ∓ Im[𝜃 + (𝑡)] + Im[𝜃 − (𝑡)] .
√
2 𝑑
(15)

From this result, the following properties of the solution
can be predicted. Note that the temporal evolution (13) of
the real part of 𝜃 ± (𝑡) has a structure similar to that of theKuramoto model, since 𝐶 ± (𝑡) > 0. The diﬀerence from theKuramoto model is that 𝐶 ± (𝑡) is not a constant. Now, if
𝐶 ± (𝑡) is large enough and phase synchronization occurs as
in a Kuramoto model, the following states are realized:
𝜋Re[𝜃 (𝑡)] + Re[𝜃 (𝑡)] = + .
2
+

−

2 𝑑

2 𝑑

Here, in order to emphasize the diﬀerence from the previous
section, the diﬀerent parts are shown in red. As in theprevious section, the two-dimensional vector of the solutionof (18) is denoted as
( + )
𝜓 (𝑡)
,
(19)
𝝍(𝑡) = −𝜓 (𝑡)and set the Ansatz of (18) as()
𝝍 ± (𝑡) := exp ∓i𝜃 ± (𝑡) ,(20)

in double sign correspondence; this yields

Along with this, the temporal change (14) of the imaginary
part of 𝜃 ± (𝑡) becomesdIm[𝜃 ± (𝑡)] = ±𝐶 ± (𝑡),d𝑡same as (6), the solution of (17) is also linked to the solutionof the wave equation (4) by (7). Therefore, there should be
no problem in adopting either (6) or (17) as the fundamental
equation.
In this section, we would like to conﬁrm whether the
echo-chamber eﬀect derived from the fundamental equation(6) in the previous section has the same result as the fundamental equation (17) in the other representation.
As in the previous section, we consider the situation thatthe degree matrix, D = 𝑑 𝑰, is proportional
to the identity√
matrix. Therefore, matrices H and D can be simultaneously diagonalized. If the fundamental equation (17) isexpressed in the block diagonalized form of 2 × 2, the equations for all eigenvalues, other than the eigenvalue of 0 of
the Laplacian matrix, are the same, and can be written as
follows:
(
[
] √ [
])
𝜔2 +1 −1d𝑑 +1 +1+
𝝍(𝑡)i 𝝍(𝑡) =√d𝑡
2 −1 −1
2 𝑑 +1 −1[ 𝜔2 +𝑑]2+ √− 𝜔 √−𝑑2𝑑2𝑑= 𝜔2 −𝑑
𝝍(𝑡),
(18)
2
+ √
− 𝜔 √+𝑑(16)

because the sin function part of (14) becomes +1. Therefore,𝜃 + (𝑡) increases and 𝜃 − (𝑡) decreases with time. In both cases,
these changes increase the amplitude of 𝜓 ± according to theAnsatz (11). This result means that the activity of the user
in the community is activated, and it is considered that thisdescribes the occurrence of the echo-chamber eﬀect.
4. Veriﬁcation of Another Representation of the Fundamental Equation
The fundamental equation (6) is one representation of thefundamental equation; another representation that satisﬁes
the anti-commutation relation (8) is given by()√
di 𝒙(𝑡)
ˆ = H ⊗ 𝒃ˆ + D ⊗ 𝒂ˆ 𝒙(𝑡),
ˆ
(17)
d𝑡
in which the 2 × 2 matrices 𝒂ˆ and 𝒃ˆ are exchanged. The

( (
))
d ±
𝜔2 + 𝑑 𝜔2 − 𝑑𝜃 (𝑡) = √ − √ exp ±i 𝜃 + (𝑡) + 𝜃 − (𝑡) .
d𝑡
2 𝑑
2 𝑑
(21)
Since 𝜃 ± (𝑡) is a complex number in general, by substituting
𝜃 ± (𝑡) = Re[𝜃 ± (𝑡)] + i Im[𝜃 ± (𝑡)]into (21), we obtain the temporal evolutions of the real andthe imaginary parts of 𝜃 ± (𝑡) asdRe[𝜃 ± (𝑡)]d𝑡()
𝜔2 + 𝑑= √ − 𝐶 ± (𝑡) cos Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)]2 𝑑
𝜔2 + 𝑑
= √
2 𝑑
( (
)
𝜋)
+ 𝐶 ± (𝑡) sin − Re[𝜃 ∓ (𝑡)] +− Re[𝜃 ± (𝑡)] ,
2
(22)
(
)
d
Im[𝜃 ± (𝑡)] = ∓𝐶 ± (𝑡) sin Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)] ,d𝑡(23)

2020 International Conference on Emerging Technologies for Communications (ICETC 2020)
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where 𝐶 ± (𝑡) is deﬁned as (15), the same as in the previous
section.
From this result, the following properties of the solution
can be predicted. Note that the temporal evolution (13) of
the real part of 𝜃 ± (𝑡) also has a structure similar to that of
the Kuramoto model. Now, if phase synchronization occurs
as in a Kuramoto model, the following states are realized:Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)] = −𝜋2This is diﬀerent from the previous section. Along with this,the temporal change (23) of the imaginary part of 𝜃 ± (𝑡)becomes()
dIm[𝜃 ± (𝑡)] = ∓𝐶 ± (𝑡) sin Re[𝜃 + (𝑡)] + Re[𝜃 − (𝑡)]d𝑡= ∓𝐶 ± (𝑡) × (−1)= ±𝐶 ± (𝑡),(24)because the sin function part of (23) becomes −1. This result
is the same as (16), and it can be seen that the behavior of
the solution does not depend on the representation of thefundamental equation.
5. Conclusion
In the oscillation model, the fundamental equation can describe user dynamics, and at the same time, describe thecausal relationship between the network structure and theuser dynamics. Since the fundamental equation is based on
an algebraic structure that satisﬁes the anti-commutation relation and its matrix representation is not unique, there are
two diﬀerent representations in the fundamental equation.
The question is, do the diﬀerent representations describe thesame user dynamics for the same phenomenon. In otherwords, if diﬀerent results are derived from the fundamental
equations, the properties of the solutions that appear will
diﬀer depending on the representations of the fundamental
equation, which means that the theoretical model will contain contradictions. In this paper, we compared the user
dynamics derived from both fundamental equations for the
echo-chamber eﬀect and conﬁrmed that they yielded thesame user dynamics.
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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