2020 International Conference on Emerging Technologies for Communications (ICETC 2020)1
Modeling of Online Echo-Chamber Effect Based on the Concept ofSpontaneous Symmetry Breaking
Masaki AIDA†a), Fellow and Ayako HASHIZUME††b), Nonmember
SUMMARY The online echo-chamber effect is a phenomenon in whichbeliefs that are far from common sense are strengthened within relativelysmall communities formed within online social networks. Since it is signif-icantly degrading social activities in the real world, we should understandhow the echo-chamber effect arises in an engineering framework to realizecountermeasure technologies. This paper proposes a model of the onlineecho-chamber effect by introducing the concept of spontaneous symmetrybreaking to the oscillation model framework used for describing online userdynamics.key words: oscillation model, online echo-chamber effect, anti-commutation relation, spontaneous symmetry breaking
1. Introduction
With the development of information networks, user activi-ties on online social networks (OSNs) using social network-ing services (SNSs) are becoming more active. While OSNscan streamline people’s activities in the real world, they canalso cause social problems such as the online flaming phe-nomenon and online echo-chamber effect. Therefore, under-standing user dynamics in OSNs is a crucial issue.
The oscillation model is known to be able to describeuser dynamics in OSNs [1], [2]. This is a minimal modelthat applies the wave equation on networks. In the oscil-lation model concept, user activities influence each otherthrough the OSN, and the wave equation models the propa-gation of their influence through the OSN at a finite speed.Since the oscillation model well models the online flam-ing phenomenon, this paper focuses on modeling the onlineecho-chamber effect.
The online echo-chamber effect is a phenomenon inwhich beliefs that are far from common sense are strength-ened within relatively small communities that develop inOSNs. This begins with forming a closed cluster of users,with whom they become aligned on a particular opinion.Such a phenomenon is similar to the concept of spontaneoussymmetry breaking (SSB) [3]. SSB refers to a situation inwhich an asymmetric state emerges that is stable due to thesymmetry originally possessed by the system breaking spon-taneously. SSB is a theoretical model that has been used todescribe spontaneous magnetization (see Fig. 1). The atomsof ferromagnetic material are small magnets, and no overallmagnetism is exhibited if each atom is oriented in a random
†The author is with Tokyo Metropolitan University, Hino,Tokyo, 191-0065 Japan.
††The author is with Hosei University, Machida, Tokyo, 194-0298 Japan.
Fig. 1 Image of spontaneous magnetization of ferromagnetic material
direction. However, if the atoms’ directions become alignedfor some reason, the material acts as a single large magnet.
Since SSB is essentially a theoretical model of quantumtheory, there is a problem in applying SSB as is to modelingsocial phenomena. Figure 2 shows the changes in the poten-tial function before and after SSB. Before SSB, it is stableat the origin. When SSB occurs, it changes to the shapein the right called the Mexican hat potential. The originbecomes unstable and stabilizes at the valley bottom in anarbitrarily selected direction. This corresponds to the phe-nomenon that the ferromagnetic atoms spontaneously alignin a specific direction. After SSB, new dynamics that orbitthe valley bottom, which did not exist before SSB, can nowexist; this is called the Nambu-Goldstone mode (N-G mode).The superficial analogy provides no understanding of howthe N-G mode corresponds to the attributes of actual OSNs.
Since the fundamental equation of user dynamics de-rived from the oscillation model has a structure similar tothe Dirac equation of relativistic quantum mechanics, SSBcan be expected to be naturally incorporated into the frame-work of the oscillation model. However, it is very artificialto introduce the Mexican hat potential function a priori; it isnecessary to identify the OSN structural change that worksin the same way as the Mexican hat potential. Also, it is nec-
(a) Before symmetry breaking (b) After symmetry breakingFig. 2 The shapes of the potential function before and after SSB
2020 International Conference on Emerging Technologies for Communications (ICETC 2020)2
essary to elucidate the emergence of the N-G mode in OSNs.This paper discusses a model of the online echo-chamber ef-fect that applies the concept of SSB to the framework of theoscillation model.
2. Oscillation Model for User Dynamics in OSNs
This section briefly summarizes the fundamental equationsof the oscillation model. First, for nodes i, j ∈ V of directedgraph G(V,E), which represents the structure of an OSNwith n users, if the weight of directed link (i → j) ∈ Eis given as wij , the adjacent matrix A = [Aij ]1≤i,j≤n isdefined as
Aij :=
{wij , (i→ j) ∈ E,0, (i→ j) ∈ E.
(1)
Also, given nodal (weighted) out-degree di :=∑
j∈∂i wij ,the degree matrix is defined as
D := diag(d1, . . . dn). (2)
Here, ∂i denotes the set of adjacent nodes of out-links fromnode i. Next, the Laplacian matrix of the directed graphrepresenting the OSN structure is defined by
L := D −A. (3)
To eliminate situations that trigger online flaming phenom-ena, we assume that all eigenvalues of L are real.
Let the state vector of users at time t be
x(t) := t(x1(t), . . . , xn(t)),
where xi(t) (i = 1, . . . , n) denotes the user state of node iat time t. Then, the wave equation for the OSN is written as
d2
dt2x(t) = −Lx(t). (4)
Here, in addition to simply finding the solution x(t) of thewave 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 betweenOSN structure and user dynamics. To achieve this, we needto develop a first-order differential equation with respectto time (hereinafter referred to as the fundamental equa-tion) [5], [6].
Not only is the fundamental equation a first-order dif-ferential equation, but its solution must also be the solutionof the wave equation (4). The simplest way to realize this isas follows. By using the positive semi-definite matrix
√L,
which is the square root of L, we introduce the followingfundamental equation (double-sign corresponds),
±id
dtx±(t) =
√Lx±(t), (5)
where√L is uniquely determined for L. The two funda-
mental equations (5) can, by using a 2n-dimensional state
vector, be expressed as a single expression of
id
dtx(t) =
(√L⊗
[+1 00 −1
])x(t), (6)
where for x±(t) = t(x±1 (t), . . . , x±n (t)) (double-sign cor-
responds), the 2n-dimensional state vector x(t) is definedas
x(t) := x+(t)⊗(10
)+ x−(t)⊗
(01
).
Also, ⊗ denotes the Kronecker product [7]. We call (6) theboson-type fundamental equation.
The boson-type fundamental equation (6) has a techni-cal issue. Even if the Laplacian matrixL is sparse, its squareroot matrix
√L is generally a complete graph (see Fig. 3).
In ordinary large-scale social networks, it is implausible toassume a situation in which all users are directly connected.The matrix that appears in the fundamental equation mustcompletely reflect the OSN link structure (whether there isa link between OSN nodes). Therefore, the solutions of theboson-type fundamental equation (6) cannot exist in OSNsthat have structures other than complete graphs.
To avoid this problem, we introduce another fundamen-tal equation. Let us introduce a new matrix, H, as follows.
H :=√
D−1 L =√D −
√D−1 A, (7)
where√D := diag(
√d1, . . . ,
√dn). As is well-known, the
normalized 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. The semi-normalized Laplacianmatrix H is also a Laplacian matrix of a certain directedgraph, which has the same link structure as L as regards linkexistence and absence (see Fig. 3).
Using the semi-normalized Laplacian matrix H, a newfundamental equation for user dynamics can be written asfollows:
id
dtx(t) =
(H⊗ a+
√D ⊗ b
)x(t), (8)
where a and b are 2× 2 matrices defined as
a =1
2
[+1 +1−1 −1
], b =
1
2
[+1 −1+1 −1
],
and x(t) is the 2n-dimensional state vector. Solution x(t)of the original wave equation (4) can be obtained from x(t)by
x(t) = (I ⊗ (1, 1)) x(t). (9)
The fundamental equation is similar to the Dirac equa-tion of relativistic quantum mechanics, and its feature is thata and b satisfy the anti-commutation relation:
{a, b} := a b+ b a = e, a2 = b2 = o, (10)
2020 International Conference on Emerging Technologies for Communications (ICETC 2020)3
0.5
0.25
0.25
1
4
2
2
4
0.5
0.5
3
3
2
1
24
3
3
0.41
0.5
0.11
0.82
1.37
0.87
1.41
1.37
0.71
0.17
1.31
1.5
0.67
0.5
1.411.33
1
1.73
Laplacian matrix L square root matrix√L semi-normalized Laplacian matrix H
Fig. 3 An example of network structures represented by the Laplacian matrix L, the square rootmatrix
√L, and the semi-normalized Laplacian matrix H
where e denotes the 2× 2 unit matrix and o denotes the nullmatrix. Therefore, we call (8) the fermion-type fundamentalequation.
3. Echo-Chamber Phenomena
One model proposed to explain the occurrence of SSB positsthat a relatively small subnetwork corresponding to a closedcommunity detaches itself from its underlying OSN, andthe isolated subnetwork becomes a complete graph [8] (seeFig. 4).
The isolated subnetworks can oscillate around a differ-ent equilibrium point than the OSN before isolation. There-fore, a biased state is realized. In addition to the fermion-type fundamental equation’s solution, a new solution of theboson-type fundamental equation (6) can exist because theisolated subnetwork is in effect a complete graph. These twofeatures can be associated with SSB features, and the lattercan be interpreted as the N-B mode.
Here, based on the fermion-type fundamental equation(8), we show the simplest case. Let us consider the situa-tion in which the weights of all isolated subnetwork linkshave the same value, w. This is a situation in which thelink’s weight, which indicates the strength of the relation-ship between users, has increased to the limit and is saturatedbecause the discussion in the isolated community is fully ac-tivated. At this time, if the number of users in the isolatedcommunity is n, the nodal degree is the same for all nodes,and d = (n − 1)w. Also, in the corresponding Laplacianmatrix, all eigenvalues other than 0 are duplicated, and theeigenvalues are denoted as λ = ω2 = nw.
In this situation, since the degree matrix is D = d I ,the matricesH and
√D can be diagonalized simultaneously.
Therefore, the fermion-type fundamental equation (8) canbe transformed so that both the matrices H and
√D are
diagonalized. Thus, the fermion-type fundamental equationis expressed in the block diagonalized form of 2 × 2. Thetransformed equations for all the eigenvalues other than theeigenvalue of 0 of the Laplacian matrix, are the same andcan be written as follows:
id
dtψ(t) =
(ω2
2√d
[+1 +1−1 −1
]+
√d
2
[+1 −1+1 −1
])ψ(t)
=
[+ω2+d
2√d
+ω2−d2√d
−ω2−d2√d
−ω2+d2√d
]ψ(t). (11)
Here, if the two-dimensional vector of the solution of (11) isdenoted as
ψ(t) =
(ψ+(t)ψ−(t)
), (12)
and set the Ansatz of (11) as
ψ±(t) := exp(∓iθ±(t)
), (13)
in double sign correspondence; this yields
d
dtθ±(t) =
ω2 + d
2√d
+ω2 − d
2√d
exp(±i(θ+(t) + θ−(t)
)).
(14)
Since θ±(t) is a complex number in general, by substituting
θ±(t) = Re[θ±(t)] + i Im[θ±(t)]
into (14), we obtain the temporal evolutions of the real andthe imaginary parts of θ±(t) as
d
dtRe[θ±(t)]
=ω2 + d
2√d
+ C±(t) cos(Re[θ+(t)] + Re[θ−(t)]
)=ω2 + d
2√d
+ C±(t) sin(−(Re[θ∓(t)]− π
2
)− Re[θ±(t)]
),
(15)d
dtIm[θ±(t)] = ±C±(t) sin
(Re[θ+(t)] + Re[θ−(t)]
),
(16)
where
2020 International Conference on Emerging Technologies for Communications (ICETC 2020)4
Fig. 4 Structural change in OSNs that yields an isolated cluster
C±(t) :=ω2 − d
2√d
exp(∓(Im[θ+(t)] + Im[θ−(t)]
)).
(17)
From this result, the following properties of the solutioncan be predicted. Note that the temporal evolution (15) ofthe real part of θ±(t) has a structure similar to that of theKuramoto model [9], since C±(t) > 0. The difference fromthe Kuramoto model is that C±(t) is not a constant. Now, ifC±(t) is large enough and phase synchronization occurs asin a Kuramoto model, the following states are realized:
Re[θ+(t)] + Re[θ−(t)] = +π
2.
Along with this, the temporal change (16) of the imaginarypart of θ±(t) becomes
d
dtIm[θ±(t)] = ±C±(t), (18)
because the sin function part of (16) 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 (13). That is since the amplitude of the Ansatz (13)is denoted as
|ψ±(t)| = exp(±Im[θ±(t)]
),
the amplitude |ψ+(t)| increases if Im[θ+(t)] increases andthe amplitude |ψ−(t)| increases if Im[θ−(t)] decreases. Thisresult means that user activity in the isolated community isactivated, and it is considered that this describes the occur-rence of the online echo-chamber effect.
4. Conclusion
This paper has proposed a model that introduces the conceptof SSB to explain the online echo chamber phenomenon.SSB originated as a theoretical model in quantum mechan-ics to explain the spontaneous magnetization of ferromag-netism. Still, an analogy with a phenomenon in whichopinions are aligned in social communities has also beendiscussed. However, superficial analogies failed to iden-tify the correspondence between social phenomenon and theNambu-Goldstone mode, new dynamics that appear when
SSB occurs. This paper has shown how the OSN struc-tural changes that reflect the user dynamics of OSN corre-spond to spontaneous symmetry breaking in the frameworkof the oscillation model; we can now explain the emergenceof new dynamics corresponding to the Nambu-Goldstonemode. This paper focused on the simplest case of user dy-namics for an isolated community based on this framework.It showed the possibility of a phenomenon in which thestrength of user dynamics increases well mirrors the onlineecho-chamber effect.
Acknowledgement
This research was supported by Grant-in-Aid for ScientificResearch 19H04096 and 20H04179 from the Japan Societyfor the Promotion of Science (JSPS).
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