NOLTA, IEICE
Paper

Increase of low-frequency modes of user
dynamics in online social networks during
overheating of discussions
Masaki Aida 1 a) , Kouichi Nagatani 2 b) , and Chisa Takano 2 c)
1

Graduate School of Systems Design, Tokyo Metropolitan University,
6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan

2

Graduate School of Information Sciences, Hiroshima City University,
3-4-1 Ozuka-Higashi, Hiroshima 731-3194, Japana)aida@tmu.ac.jp
ksts0826lv@gmail.com
c)takano@hiroshima-cu.ac.jpb)Received August 2, 2021; Revised November 5, 2021; Published April 1, 2022
Abstract: User dynamics in online social networks have a signiﬁcant impact not only on
the online community but also on real-world activities. Therefore, understanding online user
dynamics has become a signiﬁcant issue. The wave equation-based model for online social networks (called the oscillation model) is a theoretical model proposed to describe user dynamics
in online social networks. This model can be used to understand the relationship between
explosive user dynamics and the structure of social networks. However, since the oscillation
model was introduced as a purely theoretical model of social networks, it is necessary to conﬁrm whether the model describes real phenomena correctly or not. In this paper, we ﬁrst showa prediction from the oscillation model; the low-frequency oscillation mode of user dynamics
will be dominant when the structure of online social networks changes so that user dynamics
is activated. To verify the predictions with actual data, we show spectral analyses of both thelog data of posts on an electronic bulletin board site and the frequency data of word searchfrom Google Trends. The results support the predictions from the theoretical model.
Key Words: online social networks, oscillation model, user dynamics, wave equation

1. Introduction
As information networks penetrating everyday life, it has dramatically increased the exchange of
information among individuals, and the user dynamics of online social networks (OSNs) are beginningto have a major impact not only on online communities but also on social activities in the real world.
In particular, explosive user dynamics such as online ﬂaming is often seen in OSNs and can be grownmuch faster than human rational decision making can respond, which can cause major social unrest.
Here, online ﬂaming means the phenomena that the activity of user dynamics becomes extremely

511
Nonlinear Theory and Its Applications, IEICE, vol. 13, no. 2, pp. 511–532c
IEICE
2022

DOI: 10.1587/nolta.13.511

high due to the impact of user networking. Therefore understanding the user dynamics of OSNs basedon a mathematical model is an urgent and important issue. This paper focuses on the behaviors ofuser dynamics.
Online ﬂaming is generally taken to be a sudden ﬂood of hostile messages targeting the same victim.
To fully understand ﬂaming, it may be necessary to consider the background of the incident from theperspective of sociology and behavioral psychology, because ﬂaming is caused by people’s thoughts andactivities. Unfortunately, this consideration is just a post-incident coping approach. To establish atimely countermeasure to ﬂaming, a mathematical model that can describe ﬂaming without analyzing
the behavior of people is necessary.
Let us consider the question of whether we can take measures against ﬂaming without analyzing
the behavior of people. As an example, consider the bandwidth design of the classical telephone
network. Although telephone calls are generated from the activities of individual human beings, wehave successfully modeled them as random stochastic processes without modeling human behavior. If
this example is followed, it may be possible to understand online ﬂaming in some kind of mathematicalframework rather than by understanding human behavior. If successful, it will be the basis fordeveloping timely countermeasures to ﬂaming.
Online user dynamics, including explosive user dynamics, are generated by interactions between
users. It is diﬃcult to fully understand the details of interactions between actual users, but we canapply the concept of the minimal model; it models the simple interactions commonly included in many
diverse types of user interactions. The minimal model was used to develop the oscillation model on
networks to describe online user dynamics [1]. In the oscillation model approach, network dynamics
is described by the wave equation on networks.
The reason why the wave equation on networks is applied in the oscillation model describing online
user dynamics is as follows. OSN users interact and have some inﬂuence on each other. Such inﬂuences
should be propagated through the relationship of users, in other words, through OSNs. Figure 1
illusrates such propagations of inﬂuences. The propagation of inﬂuence never occurs at an inﬁnite
speed; it would occur at ﬁnite speeds.
In general, the wave equation describes the propagation of something through a medium. In ourcase, it is the inﬂuence of users propagated through OSNs. Figure 2 shows the situation that someinﬂuence of a user propagates at ﬁnite speeds from left to right. Since the waveform of inﬂuencecan be decomposed into Fourier modes, each Fourier mode should propagate at ﬁnite speeds if theinﬂuence propagates at ﬁnite speeds.
There is signiﬁcant a diﬀerence between the oscillation model for OSNs and the wave equation
on networked objects in the real world. Since the strength of inﬂuence between users is asymmetry,
Newton’s third law about action and reaction is violated in OSNs in general, unlike physical models
in the real world. This causes the violation of the energy conservation law in OSNs and causes online
ﬂaming as a result.
The validity of the oscillation model has some theoretical support because it provides a naturalderivation of conventional node centrality [2, 3]. There are the degree centrality and the betweenness

Fig. 1. Propagation of inﬂuence between users at ﬁnite speeds (node: user,
link: the relationship between users).

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Fig. 2.

Propagation and the wave equation.

centrality [4–8] as typical node centrality. The oscillation energy of each node calculated from theoscillation model gives both the conventional node centralities in special cases. In addition, theoscillation energy gives a generalization of node centrality. In some conditions, the oscillation model
can describe the characteristic of oscillation energy divergence in certain OSN conﬁgurations. Thismeans the emergence of explosive user dynamics can be explained by OSN structure [1, 9, 10].
As mentioned above, the oscillation model is simple but its theoretical framework has the potential
to be able to describe explosive user dynamics. However, the oscillation model has yet to be validated
by using actual data to conﬁrm that it can describe real phenomena correctly. In this paper, we ﬁrst
derive a prediction from the oscillation model; the low-frequency oscillation mode of user dynamics will
be dominant when the sudden increase of user activity in OSN is triggered. To verify the predictions
against actual data, we conduct spectral analyses of both the log data of posts on an electronic bulletinboard site and the frequency data of word search from Google Trends. This paper is an extended
version of the research presented in [11, 12]. The previous studies introduced some experimental results
but there were some problems; the reason behind the selection of the experimental data was unclear,and there were no comparisons with control groups in which user dynamics were not triggered. This
paper provides experimental results to rectify the omissions of those papers.
The rest of this paper is organized as follows. Sect. 2 shows related work to clarify the position of
the oscillation model. Sect. 3 brieﬂy summarizes the oscillation model. Sect. 4 shows the predictionthat the low-frequency mode of user dynamics becomes prominent based on the oscillation model.
Sect. 5 shows experimental results and clariﬁes that they do not contradict the prediction of theoscillation model. Finally, we conclude discussions in Sect. 6.

2. Related work
Spectral graph theory is one of the key approaches for investigating the structure of networks and/or
dynamics on networks, and the eigenvalues and the eigenvectors of the Laplacian matrix play important roles in investigating network structure and/or dynamics. One of the most signiﬁcant propertiesof spectral graph theory is the fact that we can introduce graph Fourier transformation [13], which isthe diagonalization of the Laplacian matrix.
Graph Fourier transformation has accelerated the development of graph signal processing [14–16], which is a generalized signal processing for graph-structured data. Graph signal processing can
generate a transformation that yields both multi-resolution views of graphs and an associated multiresolution of a graph signal [17–19]. This is applicable to multi-layer clustering for social networks.
Recently, graph neural networks (GNNs) have been actively developed [20]. This is a combination of513

the graph signal processing and deep learning technology. In particular, graph convolutional networks(GCNs) are a signiﬁcant topic in GNNs. First introduced in [18], convolution for graph signals is basedon graph Fourier transformation. As an application of GNNs to OSNs, [21] discusses a mechanism
for social recommendations.
In this paper, we are interested not in deep learning for graph data but in user dynamics in OSNs,
that is, the dynamic aspect of user activities in OSNs.
Studies on user dynamics in OSNs have examined various models in an eﬀort to capture the diversityof the characteristics of user dynamics. User dynamics that describe the adoption and abandonmentof a particular social networking service (SNS) have been modeled by the SIR model, which is a
traditional epidemiological model, and the irSIR model, which is an extension of that model [22–24].
These models express the state transition of an objective system in a macroscopic framework, but are
not good at describing individual user dynamics. Also, these models describe the speed of changes
in transient states and/or the conﬁguration of the ﬁnal steady-state of the system, but they fail toaddress the divergence of user dynamics.
The consensus problem including user opinion formation is typical of the dynamics in OSNs [25, 26].
This can be modeled by a ﬁrst-order diﬀerential equation with respect to time by using a Laplacianmatrix that represents the social network structure. The diﬀerential equation used in this model
is a sort of a continuous-time Markov chain on the network. First-order diﬀerential equations withrespect to time are also used in modeling of the temporal change in social network structure (linking
or delinking of the nodes), and there are models that consider the change via a continuous-timeMarkov chain [27]. In addition to theoretical models, [28] and [29] studied user dynamics analysis
based on real network observations. Similar to epidemiological models, the Markov chain describes
the transient states and/or the steady-state of the system, but not the divergence of user dynamics.
The signiﬁcant advantage of the oscillation model based on the wave equation in OSNs is that it
can describe the process by which the strength of user dynamics diverges.

3. Oscillation model of online user dynamics
The user dynamics in OSNs are complex, and understanding them completely is as diﬃcult as understanding human thoughts and human interactions. For this reason, we give upset aside tacklingthe complete description of user dynamics and aim at the simplest model possible that can describe
the properties that are common to many types of user interaction. This approach is called a minimalmodel. As a minimal model, the oscillation model was proposed to describe user dynamics in OSNs
and takes the form of a wave equation on networks.

3.1 Wave equation on networks
An outline of the oscillation model is summarized as follows [1].
Let an OSN be modeled by a directed graph G(V, E) with n nodes, where V = {1, . . . , n} is theset of nodes and E is the set of directed links. For nodes i, j ∈ V , the weight of the directed link(i → j) ∈ E is wij (> 0), The adjacency matrix A = [Aij ]1≤i,j≤n of the OSN is deﬁned aswij ,(i → j) ∈ E,
Aij :=(1)
0,(i → j) ∈ E.
Also, for the nodal (weighted) out-degree di := j∈∂i wij , the degree matrix is deﬁned as D :=diag(d1 , . . . dn ). Here, ∂i denotes the set of adjacent nodes of out-links from node i. Then, theLaplacian matrix of the directed graph representing the structure of the OSN is deﬁned by L := D−A.
An example of the Laplacian matrix is shown in Fig. 3.
Let the state vector of users at time t be x(t) := (x1 (t), . . . , xn (t))T , where xi (t) (i = 1, . . . , n)denotes the user state of node i at time t. Then, the wave equation on the OSN is written asd2x(t) = −L x(t).
dt2

514

(2)

Fig. 3.

An example of the Laplacian matrix.

In general, the wave equation describes the propagation of something through a medium at ﬁnite
speeds. In this case, the wave equation (2) describes that the inﬂuence between users propagates
through OSN at ﬁnite speeds. Since no inﬂuence can propagate at an inﬁnite speed in general, thewave equation-based model is a natural way of thinking.

3.2 Oscillation energy and node centrality
Although the oscillation model is simple, it has been conﬁrmed that it has the following advantages.
• The oscillation energy for each node calculated from the oscillation model not only provides
a common framework for the conventional indices of node centrality (degree centrality andbetweenness centrality [6–8]) but also can provide an extended concept of node centrality thatis applicable to OSNs with more complex conﬁgurations [2, 3].
• it is possible to describe the phenomenon in which the activity of user dynamics grows explosively
as seen in the online ﬂaming phenomenon in OSNs [1].
To explain the above advantages brieﬂy, we ﬁrst deﬁne here the symmetrizable directed graph.
Since the row sum of the Laplacian matrix L is zero, one of the eigenvalues of L is 0. Thus, we letm = (m1 , . . . , mn )T be the left eigenvector associated with the eigenvalue of 0, that is,
mT L = (0, . . . , 0).
The directed graph is symmetrizable if and only if there is an m all of whose components are positivemi > 0 (i = 1, . . . , n) and satisfy
mi wij = mj wji ,for all adjacent nodes i and j. Hereafter, the Laplacian matrix for a symmetrizable directed graph isdenoted as L0 .
Let M be a diagonal matrix, M := diag(m1 , . . . , mn ), whose diagonal components are the components of m. The Laplacian matrix L0 for any symmetrizable directed graph can be written as the
product of M −1 and a (symmetric) Laplacian matrix L for a certain undirected graph, as follows;
L0 = M −1 L.

(3)An example of this decomposition is shown in Fig. 4.
The wave equation (2) for symmetrizable directed graph L0 can be written asd2x(t) = −L0 x(t).
dt2Fig. 4. An example of the symmetrization of the Laplacian matrix for a
directed graph.

515

(4)

Fig. 5.

Simple example for 1-dimensional network.

Rewriting the wave equation (4) using (3) givesMd2x(t) = −L x(t).
dt2

From this, it can be seen that M corresponds to the masses of nodes and the link weights of theundirected graph represented by L corresponds to the spring coeﬃcients. Figure 5 illustrates anexample situation using a simple 1-dimensional network. Here, we introduce the scaled Laplacianmatrix S0 for symmetrizable directed graph L0 as
S0 := M +1/2 L0 M −1/2 ,

(5)

√
√
where M +1/2 := diag( m1 , . . . , mn ). For symmetrizable directed graph L0 , scaled Laplacianmatrix S0 is a real symmetric matrix. The wave equation (4) can, by using S0 , be written asd2y(t) = −S0 y(t),
dt2

(6)

where y(t) := M +1/2 x(t).
To solve the wave equation (4), we solve (6) ﬁrst. Let us deﬁne the eigenvalues and eigenvectorsof S0 . Since S0 is a real symmetric matrix, all the eigenvalues are real numbers. In addition, theminimum value of the eigenvalue is 0. Thus we sort the eigenvalues as
0 = λ0 ≤ λ1 ≤ · · · ≤ λn−1 ,where the multiplicity of eigenvalues of 0 is the number of connected components of the OSN. Let the
eigenvector associated with eigenvalue λμ be vμ . Since S0 is a real symmetric matrix, we can choose
eigenvectors {vμ }μ=0, 1, ..., n−1 as the orthonormal eigenbasis. That is, eigenvectors are mutuallyorthogonalized and their lengths are 1;vμ · vν = δμν ,where δμν is the Kronecker delta, that is, δμν = 1 if μ = ν, and δμν = 0 if μ = ν. S0 is obtained by
the similarity transformation (5) of L0 . It follows that the eigenvalues of S0 are the same as those of
L0 .
The solution of (6) can be expanded by the eigenbasis to yield
y(t) =n−1

aμ (t) vμ .

μ=0

By substituting this into (6), we have the following n independent equationsd2
aμ (t) = −λμ aμ (t),
dt2

516

(7)

for the oscillation modes μ = 0, 1, . . . , n − 1. Each of these equations is the equation of the motionof a simple harmonic oscillator. The solution of (7) is obtained aswhere ωμ :=as

−aμ (t) = c+
μ exp(+iωμ t) + cμ exp(−iωμ t),(8)−
λμ is the eigenfrequency, and c+
μ and cμ are constants. The solution of (4) is obtainedx(t) =

n−1

μ=0

 +
 −1/2cμ exp(+iωμ t) + c−vμ .
μ exp(−iωμ t) M

(9)From the solution (9), the oscillation energy Ei of node i for symmetrizable directed graphs isobtained as
Ei =

n−1

μ=0


2
− 2
λμ |c+vμ (i)2 ,μ | + |cμ |(10)where vμ (i) is the i-th component of the eigenvector vμ = (vμ (1), . . . , vμ (n))T . The oscillation energy
for each node gives the conventional node centrality (degree centrality and betweenness centrality) as
special cases [2, 3]. Also, as the oscillation energy gives a generalized notion of node centrality.
In general, the Laplacian matrix L of OSNs is not necessarily a symmetrizable directed graph. In
such situations, the wave equation on network does not correspond to physical dynamics like shown as
Fig. 5. Interaction between adjacent nodes violates Newton’s third law and the energy conservation
law of the system does not hold, in general.
For a general directed graph, the eigenvalue of the Laplacian matrix is not restricted to real numbers.
Even in that case, the real part of the eigenvalues is known to be non-negative. From the Gershgorin
circle theorem [9, 30], all the eigenvalues are included in the largest Gershgorin disk on the complex
plane. Here, the largest Gershgorin disk of the Laplacian matrix is a circle whose center is on the real
axis at the maximum weighted nodal degreedmax = max di ,i∈Vand its radius is also dmax (see Fig. 6).
If eigenvalue λμ of the Laplacian matrix L for an OSN is not a real number, eigenfrequency ωμ is
also a non-real number. Let the non-real eigenfrequency ωμ ∈ C be
ωμ = a + bi,
where a, b ∈ R and a ≥ 0. Note that ωμ2 = λμ .
In this case, the eigenvectors {vμ }μ=0, 1, ..., n−1 associated with eigenvalue λμ are no longer mutuallyorthogonalized. However, we can assume that eigenvectors {vμ }μ=0, 1, ..., n−1 are linearly independent.
Thus we can solve the wave equation (2) similarly and the solution for oscillation mode μ is obtainedas (8). As the eigenfrequency is not real, the solution for oscillation mode μ is written as

Fig. 6.

The maximum Gershgorin disk of L.

517

−aμ (t) = c+
μ exp(+i(a + bi)t) + cμ exp(−i(a + bi)t)−= c+
μ exp(−bt) exp(+iat) + cμ exp(+bt) exp(−iat).

This means that the amplitude of aμ (t) diverges with time as exp(+|b|t), which causes the divergenceof the oscillation energy. We consider that this process underlies the explosive user dynamics generated
in OSNs including online ﬂaming.
Let Etotal be the oscillation energy for the whole network. In the oscillation model, Etotal represents
the strength of the users’ activity, and it is an observable quantity, whereas the users’ state, x(t), is
not observable. As mentioned earlier, x(t) is given as a solution to the wave equation, and the wave
equation is required for describing the propagation of the inﬂuence between users at ﬁnite speeds.
This structure is similar to quantum mechanics. Even in quantum mechanics, the wave function that
represents the state of the system is a solution of a certain wave equation, that cannot be observed
itself. What can be observed is the square of the absolute value of the wave function corresponding
to the oscillation energy. In Sect. 5, we will verify the prediction of the oscillation model withexperimental data, but in that case, the observable amount of the oscillation energy is the subject ofanalysis.

4. Increase of low-frequency mode in the oscillation model
The oscillation model is derived from a purely theoretical framework. Although it is possible to form
the basis of the conventional node centrality from the oscillation model, it is not yet clear whether it
adequately describes the actual user dynamics of OSNs. This section explains the characteristics that
increase the strength of the low-frequency mode predicted by the oscillation model in the situationthat online user dynamics becomes highly active; the predicted characteristics are veriﬁed by assessing
actual data.

4.1 Decomposition of the Laplacian Matrix
Since the structure of OSNs is not symmetrical directed graphs in general, we should consider theoscillation model for general directed graphs. We now consider how to decompose the Laplacian matrix
of a directed graph into two parts: one is the Laplacian matrix L0 for a symmetrizable directed graphand the other is the Laplacian matrix for a one-way link graph, LI , that is,
L = L 0 + LI ,(11)

where the one-way link graph has only one-way directed links between nodes.
Figure 7 shows examples of decomposition (11) of a directed graph. As shown in the ﬁrst decomposition of Fig. 7, the directed graph on the left-hand side is composed of the symmetrizable directed
graph depicted in Fig. 3 and a one-way link graph. The second decomposition of Fig. 7 yields anundirected graph and a one-way link graph. The undirected graph is a special case of symmetrizable
directed graphs; M = I (unit matrix) in (3). The link weight between node i and node j of theundirected graph is min(wij , wji ).

Fig. 7.

Examples of the decomposition L = L0 + LI .

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From the above example, we can recognize the following facts:• Any non-symmetrizable directed graph can be decomposed into a symmetrizable directed graphand a one-way link graph.
• The decomposition is not unique, as it depends on the choice of symmetrizable directed graphincluded in the original directed graph.
If the original directed graph is symmetrizable, we can choose a decomposition such that LI = O(the null matrix). The variety of decompositions (11) of the Laplacian matrix links to the selectionof orthogonal basis in the solution space through the selection of symmetrizable directed graphs [1,9]. Choosing an appropriate coordinate system is important for a clear understanding of the analysis
target.

4.2 Online flaming and critical phenomena
From (10), the oscillation energy is time-independent for symmetrizable directed graphs, so no ﬂaming
can occur in any symmetrizable directed graph. Let us consider a general directed graph by adding
a one-way link graph to a symmetrizable directed graph. Here, let the strength of the inﬂuence of
one-way link graphs be indicated by a parameter,  ≥ 0, and we introduce Laplacian matrix L() for
general directed graphs as
L() = L0 +  LI .

(12)

The value of  indicates the magnitude of deviation from the symmetrizable directed graph, andL(0) = L0 .
It is worth noting that we do not assume the structure of OSN should be a symmetrizable graph.
Our purpose here is to investigate the diﬀerence in user dynamics depending on the OSN structures.
Here, the structure is parametrized by  and the value of  is used as an index representing the deviationfrom a symmetrizable graph. In the rest of this subsection, we show a preliminary evaluation of user
dynamics using an example network model.
As an example, we consider the following network model. The symmetrizable directed graph isgiven as⎡⎤ ⎡11
−3 −10/3 −5/3 −3
3⎢−9/4 23/4 −5/4⎥⎢
0
−9/4⎥ ⎢0⎢⎢⎥ ⎢L0 = ⎢ −10−523
0−8 ⎥ = ⎢0
⎢
⎥ ⎢
⎣−5/2
0
0
11/2
−3 ⎦ ⎣0−9/4 −9/4
−2
−3/2
8
0

0
4
0
0
0

0
0
1
0
0

0
0
0
2
0

⎤−1 ⎡
0
33 −9 −10 −5⎥⎢0⎥ ⎢ −9 23 −50⎥ ⎢0⎥ ⎢−10 −5 230⎥ ⎢0⎦ ⎣ −500114−9 −9 −8 −6

⎤
−9
−9⎥
⎥
⎥
−8⎥ ,
⎥
−6⎦
32

and the one-way link graph is given as
⎡

⎤
1
0
0
0 −1
⎢ 0
2 −1 −1 0 ⎥⎢⎥⎢⎥LI = ⎢ 001
0 −1⎥ .
⎢
⎥
⎣−1 0
0
1
0⎦
0 −1 0
0
1

Figure 8 illustrates them. The left panel of Fig. 8 shows the symmetrizable directed graph depictedin the symmetrization manner similar that used in Fig. 4 and the right panel shows the one-way linkgraph.
We demonstrate a typical example of user dynamics by using this network model. Figure 9 shows
the time variation of user state x1 (t) for node 1 given the  = 0, 1.5, 1.65 and 1.66. The initial vector
is set to x(0) = (10, 2, 7, 5, 6)T . The boundary value of the parameter at which non-real eigenvalues
appear lies in the range of 1.65 <  < 1.66. For  > 0, the network model is not symmetrizable but all
the eigenvalues are real for  < 1.65. For  > 1.66, non-real eigenvalues cause divergence. From these519

Fig. 8. Directed network model composed of a symmetrizable directed graphand a one-way link graph.

Fig. 9. Time variation of displacement x1 (t) of node 1 for directed graphs ofvarious values of .

results, it can be seen that as the value of  increases, low-frequency beats occur and grow, before
non-real eigenvalues appear. This phenomenon is not limited to the above example. The reason forthis is explained in the next subsection.
It is natural to think that OSN structure changes over time. The situation being discussed hereis that changing the structure of the OSN by increasing the value of  causes the change of the user
dynamics to be activated. So, even if it does not lead to online ﬂaming, we can expect low-frequency
beats to appear when online user dynamics are activated.
Note that symmetrizable directed networks never diverge, but its converse is not always true. For
0 <  < 1.65, the OSN structure is not symmetrizable but all the eigenvalues are real and no divergenceoccurs. Moreover, although actual OSNs should exert some damping eﬀect on user dynamics (that is,
the eﬀect of calming user dynamics over time) [1, 31], the numerical example described here does notconsider this eﬀect, for simplicity of explanation. If the strength of the damping eﬀect is suﬃciently
large, user dynamics do not diverge even if some eigenvalues of OSN become non-real.

4.3 Why low-frequency mode appears
Section 4.2 shows preliminary evaluations by using a simple network model as an example. However,
the occurrence and growth of the low-frequency beats can be explained as general phenomena basedon the framework of the oscillation model. Let us consider the characteristic function of Laplacianmatrix L() for ;det(L() − λ I) = 0.
For  = 0, since L(0) = L0 is a symmetrizable directed graph, all the eigenvalues are real. The upperleft panel of Fig. 10 shows this situation. The solutions of the characteristic equation correspond520

Fig. 10. Changes in the shape of the characteristic polynomial det(L()−λ I)
when  is increased: Two eigenvalues approach each other before the solutionsof the characteristic equation become non-real numbers.

to the intersections with the horizontal axis. Changes in the shape of the characteristic polynomial
det(L() − λ I) when  is increased are schematically displayed in the order of the arrows. The laststate is where the intersection with the horizontal axis has disappeared and non-real eigenvalues haveappeared. Before reaching that state, two eigenvalues begin to approach each other on the horizontal
axis; they disappear immediately after they intersect. Since an eigenfrequency is the square root of
its eigenvalue, the two eigenfrequencies approach when the corresponding eigenvalues approach. The
approach of the two eigenfrequencies triggers the low-frequency beats.
As a familiar example, let us consider the superposition of two oscillations of the angular frequencies
ω1 and ω2 . From the well-known formula of trigonometric functions, the superposition can be written
as




ω1 + ω2
ω1 − ω2t cost .
sin(ω1 t) + sin(ω2 t) = 2 sin
22
If the angular frequencies ω1 and ω2 are close to each other, ω1 − ω2 becomes a very small value. A
wave oscillating at the average angular frequency (ω1 + ω2 )/2 appears as a waveform modulated bya low-frequency wave with angular frequency (ω1 − ω2 )/2.
Next, we consider why the amplitude of the low-frequency mode grows. Since  = 0 gives a symmetrizable directed graph, the scaled Laplacian matrix S0 is a symmetric matrix and its eigenvectors{vμ }μ=0, 1, ..., n−1 can be selected as an orthonormal basis. The scaled Laplacian matrix for general
directed graphs is deﬁned asS() := M +1/2 L() M −1/2 .
For  > 0, S() is an asymmetric matrix.
Thus the eigenvectors {vμ ()}0≤μ≤n−1 of S() are not orthogonal to each other. For solution y(t)of the wave equationd2
y(t) = −S() y(t),dt2
we expand y(t) by the (non-orthogonal but linearly independent) eigenvectors asy(t) =n−1

aμ (, t) vμ ().

μ=0Here, the expansion coeﬃcients aμ (, t) are the coordinate values of the space spanned by the obliquebase vectors {vμ ()}0≤μ≤n−1 . Figure 11 schematically shows the coordinate value of each axis, whichis expanded in an orthogonal coordinate system, and the change in the coordinate value of each axis,
which is expanded on an oblique axis. As two eigenvalues approach each other as shown in Fig. 10,
the angle between the two eigenvectors associated with the two eigenvalues becomes smaller. At the
point of the two eigenvalues intersect, the corresponding eigenvectors become parallel. Although x(t)521

Fig. 11.

Expansion coeﬃcient increases as two eigenvectors become parallel.

is a point at the same position relative to the origin, the absolute value of the coordinate value aμ (, t)
increases on the oblique axes. This is why the amplitude of the low-frequency mode grows when the
eigenvalues approach each other.
The low-frequency beats appear not only in the user state xi (t) but also in the oscillation energyof the OSN. According to [9], if we choosey(t) =n−1

|aμ (, t)| e−ωµ t vμ (),μ=0

for simplicity, the oscillation energy of the whole OSN is obtained as

2
n−1
 dyμ (t) 1
Etotal = 2 μ=0 dt n−1 n−11=|aμ (, t)| |aν (, t)| ωμ ων (vμ () · vν ()) exp[ −i (ωμ − ων ) t]
2 μ=0 ν=0
n−1
n−1
 n−1

1
2 2
=
|aμ (, t)| ωμ +|aμ (, t)| |aν (, t)| ωμ ων (vμ () · vν ()) cos[ (ωμ − ων ) t] , (13)2 μ=0μ=0 ν=μ+1

where |aμ (, t)| denotes the amplitude of aμ (, t). This means that the oscillation energy is timedependent if the eigenvectors {vμ ()}0≤μ≤n−1 are not orthogonal to each other, that is vμ ()·vν () = 0
for some μ and ν. Therefore, the low-frequency beats appear also in the oscillation energy, if theeigenfrequencies approach each other.
Here, we consider the spectrum in the low-frequency beat of the oscillation energy. For simplicity,
let us consider two triangular waves cos(ωt) with close angular frequencies: ω = 0.10 and ω = 0.11.
The left panels in Fig. 12(a) and (b) denotes them, respectively. Also, their spectral distributions areshown in the right panels in Fig. 12(a) and (b), where the spectral distribution is normalized thatthe total volume is 1. Here, the horizontal axes in the right panels denote the frequency f and it
is deﬁned as f := ω/(2π) × 4096. The peaks of the spectral distribution appear at f = 65.2 and
f = 71.7, respectively. The sum of the two triangular waves is shown in Fig. 12(c). A low-frequency
beat appears in the left panel, but the spectral distribution has two peaks of individual waves and nolow-frequency mode.
In the oscillation model, the strength of user activity is obtained as the oscillation energy. The
oscillation energy is related to the square of waves. Figure 12(d) shows the square of the sum of thetwo triangular waves and their spectral distribution. The spectral distribution includes mainly two
oscillation modes: higher frequency around f = 137.0 = 65.2 + 71.7 and lower frequency around f =6.5 = |65.2 − 71.7|. Here, the lower frequency is the low-frequency beat. Moreover, Fig. 12(e) showsa moving average of the square of the sum of the two triangular waves and its spectral distribution.
The size of the window for the moving average is 64. The meaning to take the moving average isobserving the number of speciﬁc events during a certain time period. The details are shown in thenext section. Such observation obtains the averaged value of the target in the observing time interval.
We can recognize that the low-frequency mode is clearly observed in the oscillation energy from thespectral distribution.

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Fig. 12.

Low-frequency mode appears in the observed oscillation energy.

Finally, we summarize the framework of observations of low-frequency modes in user activity in
OSNs (see Fig. 12). The oscillation model is based on the idea that the inﬂuence between users
propagates at ﬁnite speeds and gives the strength of the user’s activity as oscillation energy as anobservable quantity. The wave equation in OSNs is necessary to describe the propagation with a ﬁnite
speed. If we succeed to observe low-frequency beats in the actual user dynamics, it can be said tosupport the eﬀectiveness of the oscillation model.

5. Experimental confirmation of low-frequency mode occurrence
As shown in the previous section, the oscillation model of online user dynamics predicts low-frequency
oscillation mode appearance in the strength of user dynamics when the network structure changes
to activate user dynamics. Accordingly, conﬁrming the appearance of low-frequency modes in real
OSN data can be considered a reasonable test of the oscillation model’s validity. Figure 13 illustrates
the structure of the relationship between the validity of the oscillation model and the observationsof low-frequency mode in user dynamics. This section shows spectral analyses of real data from abulletin board system (BBS) and Google Trends to conﬁrm the appearance of low-frequency modes.
In the following analyses, online ﬂaming does not occur, but the intensity of user dynamics grows
as the discussion within the OSN change to heat up. We verify the occurrence of the low-frequency
mode in the highly active periods.

5.1 Spectral analysis of BBS data
2-Channel (the current name is 5-Channel) is the largest BBS in Japan. This subsection introduces
additional research over our previous study [11]. In the previous study, we analyzed time-series data
sequences of posts to the BBS for diﬀerent threads. Concretely, we performed the spectral analysis
by the fast Fourier transformation (FFT) in the period that the number of postings was large (heavy523

Fig. 13.

Framework of the observation of low-frequency beat.

posting period) and the period that the number of postings was small (light posting period) andcompare them. As a result, heavy posting periods exhibited stronger low-frequency modes thanlight posting periods. This result supports the theoretical predictions suggested by the oscillation
model. However, because [11] does not explain the reasoning behind the selection of threads used to
identify heavy/light posting periods, the results presented were of interest but were not deﬁnitive.
This subsection introduces further experimental results that rectify the shortcomings of [11].
We analyzed time-series data sequences of posts to the BBS for two diﬀerent threads. The ﬁrst
thread is a Japanese stock market-related thread, the second one is for the Japanese professional
baseball team “Hiroshima Toyo Carp”. The reasons for selecting these threads are as follows. Our
purpose is to compare the strength of user dynamics in periods in which the discussion becomesanimated with that in the periods of a normal state. The stock market-related thread is strongly
aﬀected by stock price ﬂuctuations. When the stock price changes signiﬁcantly, the discussion in
the thread becomes animated, but when the stock price is basically unchanged, the discussion in thethread is not lively. Therefore, it is easy to compare the panic-like heavy posting periods with thenormal light periods. Professional baseball is very popular in Japan, and the team “Hiroshima Toyo
Carp” has been in the spotlight because it won the league title for the third consecutive year. The
analysis period is the period during and after the Japan series (Japan version of the World Series),
including the period in which many fans enthusiastically participated in the discussions. This choice isalso suitable for comparing periods when user debates are animated with those of quiescent discussion.
In addition to the above threads, [11] includes a Korean economics-related thread. This is also
suitable for selecting heavy and light posting periods. However, the selection of multiple heavy andlight posting periods diﬀerent from [11] is diﬃcult, so we omit it in the following discussion.
5.1.1 Thread for Japanese stock market
Figure 14 shows the number of posts per day from November 19, 2018, to December 7, 2018, madeon the Japanese stock market-related thread. In Fig. 14, the period indicated by a blue arrow has arelatively light-postings period, while that indicated by red arrows have heavy-postings periods. The
length of each period has successive 256 data long for the number of posting per 16 min. (about 2.84
days long). In [11], only one pair of light and heavy posting periods were analyzed. Thus it waspossible the period chosen inadvertently matched the hypothesis. In this experiment, we chose as
many heavy posting periods as we could choose without duplication. Unfortunately, there is only one
light posting period due to the length of its duration; this is discussed later.
To compare the spectral distributions for the periods, we processed the data as follows. We removed
the oscillation mode with a frequency of 0 to extract only oscillating modes. Also, to compare the
distribution of the ratio of each frequency component, the data were normalized such that the sumof distribution values was 1. Figure 15 shows the normalized FFT (moving average with a window
size of 20) of the data in each period. Markers indicated in Fig. 15 correspond to those in Fig. 14.
This result shows that the heavy posting periods have stronger low-frequency modes than the light524

Fig. 14. The number of posts per day on a Japanese stock market-related
thread in 2-Channel.

Fig. 15. Moving average of oscillation mode for window size of 20 for
Japanese stock market-related thread (256 data long).

posting period.
It is worth noting that whether the average number of posts in each period is high or low does
not impact the spectral analysis. They are related to oscillation mode with a frequency of 0, butthey are excluded from the data, and the other oscillation modes are also normalized to be 1 overall.
This also applies to the subsequent spectral analysis in this paper. Moreover, the moving average was
executed after the FFT, and the window size of the moving average was selected in order to display
the diﬀerence in data in an easy-to-see manner.
In the above evaluation, only one sequence of a light posting period could be selected. To selectand evaluate multiple light posting periods, we halved the period length to create 128 contiguous 16
minute periods (total is about 1.42 days). Under this condition, we select the top 3 non-overlapped
periods from the heavy posting periods and the top 3 non-overlapped periods from the light posting
period and analyze them. Note that this shortening of the data period is disadvantageous in termsof analyzing the low-frequency region. Figure 16 shows the normalized FFT (moving average with awindow size of 20) of the data in each period. To emphasize the low-frequency region, the horizontal
axis is plotted using a log scale. This result also shows that the heavy posting periods have stronger
low-frequency modes than the light posting period.

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Fig. 16. Moving average of oscillation mode for window size of 20 for
Japanese stock market-related thread (128 data long).

5.1.2 Thread for Japanese professional baseball team “Hiroshima Toyo Carp”
Next, Fig. 17 shows the number of posts per day from October 27, 2018, to December 3, 2018, made on
the Japanese professional baseball team “Hiroshima Toyo Carp” related thread. In Fig. 17, the period
indicated by blue arrows have relatively light-postings periods (indicated by blue and black markers),
while that indicated by red arrows has a heavy-postings period (indicated by the red marker). The
length of each period has successive 256 data long for the number of posting per 16 min. (about 2.84
days long). In [11], only one pair of light and heavy posting periods were analyzed. In this experiment,
we chose two unique light posting periods.
Figure 17 shows the normalized FFT (moving average with a window size of 20) of the data in eachperiod. This result also shows that the heavy posting period has stronger low-frequency modes thanthe light posting period.

5.2 Spectral analysis of Google trends data
We spectrally analyze time-series data of user interests in words obtained from Google Trends. Google
Trends is a web service oﬀered by Google that analyzes the popularity of words in Google searches
across a certain region in a period. For various words such as news keywords attracting attention,
we extract the strong interest period and weak interest period and analyze spectral distribution in

Fig. 17. The number of posts per day on a Japanese professional baseballteam (Hiroshima Toyo Carp) related thread in 2-Channel.

526

each period. The popularity of Google search words does not directly correspond to actual discussion
among users in OSNs, but it shows how much OSN interest there is in the topic containing the word.
Therefore, it is considered that if words associated with particular topics are used in more searches,
those topics are attracting signiﬁcantly more OSN attention, and that word search frequency can beused as an index of the strength of user interest in the related topic.
The data of user interest in a particular word, obtained from Google Trends, is called the searchinterest. The speciﬁcations of the search interest are summarized as follows. The search interest is
an index that shows how much a word is searched in Google in a certain period. The data retrievablefrom Google Trends is limited to a continuous week. In addition, the values of the search interest arenormalized so that the maximum value in the time-series data is 100.
For this paper, we retrieved multiple hourly time-series data sequences of the search interest (oneweek periods) and created long-term data by fusing those one-week data sets. Consider the example
of fusing data set from January 1 to January 7, 2019, and data set from January 7 to January 13,
2019. Here, if the value of the former at 0:00 January 7 is 80 and the value of the latter at 0:00 on
January 7 is 40, the values of the former are multiplied by 0.5. This approach ensures that the largest
value in the fused data is 100. The two data sets are then seamlessly combined on the shared date.
In this paper, our target area is Japan.
This subsection details further research that extends our previous study [12]. In the previous study,
we analyzed the time-series of the search interest data for some diﬀerent words of interest. As a result,strong interest periods exhibited stronger low-frequency modes than weak interest periods. This result
supports the theoretical predictions suggested by the oscillation model. However, because [12] does
not discuss the reasons for the selection of the words, further research was deemed necessary. We areinterested in the periods in which a certain keyword suddenly attracted attention and was discussed.
Not the characteristic of daily, weekly, or monthly changes in search interest. That is, we need some
control groups whose search interests change periodically, and to verify that the low-frequency mode
does not change in these situations. This subsection introduces further experimental results that
rectify the shortcomings of [12].
For comparison, we select two words discussed in the previous study [12]. One is “7pay”: aJapanese smartphone payment service and the other is “Fukuoka Softbank Hawks”: a Japanese
professional baseball team. 7pay was an electronic payment system that attracted attention, but due
to security problems, there were frequent unauthorized accesses just after the service started, and theservice was eventually abolished. Since this incident became big news and attracted much attention,
the periods before and after the incident were analyzed. The Japanese professional baseball team,Fukuoka Softbank Hawks, defeated the aforementioned Hiroshima Toyo Carp in the Japanese series

Fig. 18. Moving average of oscillation mode for window size of 20 for a
Japanese professional baseball team (Hiroshima Toyo Carp) related thread(256 data long).

527

and became the Japanese champion. Since the next season of Fukuoka Softbank Hawks was getting
a lot of attention, the periods before and after the start of the next season were analyzed. In bothwords (and the remaining words selected in [12]), ﬂuctuations in search interest are not routine, but
strongly dependent on topical news.
In addition to the above two words, we select the following three words; “fgo”: a smartphone game,“shuukatsu”: job hunting, and “denshi kessai”: electronic payment. They exhibited periodic changes
in search interest and their trends have decreasing, increasing, and constant tendencies, respectively.
5.2.1 “7pay”: a Japanese smartphone payment service
The left panel of Fig. 19 shows the hourly transitions in the search interest for a Japanese smartphone
payment service from May 1, 2019, to July 26, 2019; it also indicates the periods analyzed. The
period indicated by black dotted lines had relatively few searches, while that indicated by red dottedlines had many searches. The same color usage is adopted for other data sequences. The right panelof Fig. 19 shows the normalized FFT (moving average with a window size of 50) of the data in eachperiod. This result shows that the period of more interest has stronger low-frequency modes.
5.2.2 “Fukuoka Softbank Hawks”: a Japanese professional baseball teamThe left panel of Fig. 20 shows hourly transitions in the search interest for the Japanese professional
baseball team “Fukuoka Softbank Hawks” from January 1, 2019, to May 21, 2019. The right panelof Fig. 20 shows the normalized FFT (moving average with a window size of 50) of the data in eachFig. 19. Hourly transitions in interest in Japanese smartphone payment service “7pay”(left) and moving average of the normalized spectral distribution
with window size of 30 (right).

Fig. 20. Hourly transitions in the interest in Japanese professional baseballteam “Fukuoka Softbank Hawks”(left) and moving average of the normalized
spectral distribution with window size of 50 (right).

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period. This result also shows that the period of more interest has stronger low-frequency modes.
5.2.3 “fgo”: a smartphone gameThe left panel of Fig. 21 shows hourly changes in the search interest for the smartphone game “fgo”
from June 1, 2019, to September 21, 2019. The right panel of Fig. 21 shows the normalized FFT(moving average with a window size of 50) of the data in each period. This result shows that all the
periods exhibit similar spectrum distributions.
5.2.4 “shuukatsu”: job hunting
The left panel of Fig. 22 shows hourly transitions in the search interest for job hunting “shuukatsu”
from May 7, 2019, to August 13, 2019. The right panel of Fig. 22 shows the normalized FFT (moving
average with a window size of 10) of the data in each period. This result also shows that all the
periods exhibit similar spectrum distributions.
5.2.5 “denshi kessai”: electronic payment
The left panel of Fig. 23 shows hourly changes in the search interest for electronic payment “denshi
kessai” from June 15, 2019, to September 18, 2019. The right panel of Fig. 23 shows the normalized
FFT (moving average with a window size of 10) of the data in each period. This result also showsthat all the periods exhibit similar spectrum distributions.
Even if there is a diﬀerence in the strength of the search interest, it can be seen that the combinationof periodic ﬂuctuations and long-term trends without sudden excitement does not change the spectral

Fig. 21. Hourly changes in the interest in smartphone game “fgo” (left) andmoving average of the normalized spectral distribution with window size of 50(right).

Fig. 22. Hourly changes in the interest in job hunting “shuukatsu” (left) andmoving average of the normalized spectral distribution with window size of 10(right).

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Fig. 23. Hourly changes in the interest in electronic payment “denshi kessai”(left) and moving average of the normalized spectral distribution with window
size of 10 (right).

distribution. With all results gathered from Google Trends, the low-frequency mode of user dynamics
increases when the topic that attracts attention emerges, but there is no change in normal ﬂuctuations.
This supports the prediction that low-frequency modes become apparent when the structure of the
OSN changes to activate online user dynamics, which is predicted from the oscillation model.

6. Conclusion
Understanding user dynamics in OSNs is of signiﬁcant interest to understand not only information
networks but also real social activities. Many approaches have been introduced to investigate user
dynamics. Most approaches used for describing the diﬀusion of information are based on the infectious
disease model, such as the SIR model. While this approach expresses the state transition of the system
in a macroscopic framework, it is not good at describing individual user dynamics. Other models use
either the diﬀusion model or the continuous-time Markov chain, both of which are based on ﬁrst-order
diﬀerential equations with respect to time. Both models are good at describing the steady-state of
the system but are inappropriate for describing explosive user dynamics.
The oscillation model is based on the wave equation in OSNs, which is a second-order diﬀerential
equation with respect to time. The remarkable features of the oscillation model include (i) it is basedon a minimal model that describes the properties shared by a wide range of user dynamics, (ii) it
gives a uniﬁed interpretation of the conventional indices of node centrality, and it also yields theextended concept of node centrality, and (iii) explosive user dynamics like ﬂaming can be described.
In addition, modeling by the wave equation is suitable for describing the situation in which theinﬂuence of interaction between users propagates through the network at finite speeds. However, since
the oscillation model was, up to now, considered in purely theoretical terms, it was necessary to
conﬁrm whether the model describes real phenomena correctly or not.
We extracted from the oscillation model the prediction that the low-frequency oscillation mode
would be dominant when the structure of OSNs changes to activate online user dynamics. We conﬁrmed the validity of the oscillation model by verifying this prediction with actual data. Our tests
included conducting spectral analyses of the log data of posts on electronic bulletin board sites andword search frequency from Google Trends. As the results of processing the data match the predictions, the oscillation model can appropriately describe the user dynamics. The results conﬁrm theprediction generated from the theory.
If the oscillation model can be veriﬁed as suitable for describing online user dynamics, a techniquefor mitigating explosive user dynamics may be realized. The oscillation model yields two strategies.
The ﬁrst is a method that changes the link structure so that all eigenvalues of the Laplacian matrix
representing the structure of the online social network are real numbers. This corresponds to operations such as delaying the time of entering a new post on the electronic bulletin board or delaying
the arrival time of email. The other is to increase the damping strength of user dynamics. This530

corresponds to the operation of quickly attenuating the degree of attention to the topic currently
drawing attention by transmitting new information diﬀerent from the current topic of interest.

Acknowledgments
This research was supported by Grant-in-Aid for Scientiﬁc Research (B) No. 19H04096 (2019–2021),
No. 20H04179 (2020–2022), and No. 21H03432 (2021–2023), and Grant-in-Aid for Challenging Research (Exploratory) No. 21K19775 (2021–2023) from the Japan Society for the Promotion of Science(JSPS), and TMU local 5G research support.

References[1] M. Aida, C. Takano, and M. Murata, “Oscillation model for describing network dynamics causedby asymmetric node interaction,” IEICE Transactions on Communications, vol. E101-B, no. 1,
pp. 123–136, 2018.
[2] C. Takano and M. Aida, “Proposal of new index for describing node centralities based on
oscillation dynamics on networks,” IEEE GLOBECOM 2016, 2016.
[3] C. Takano and M. Aida, “Revealing of the underlying mechanism of diﬀerent node centralities based on oscillation dynamics on networks,” IEICE Transactions on Communications,vol. E101-B, no. 8, pp. 1820–1832, 2018.
[4] L.C. Freeman, “Centrality in social networks: Conceptual clariﬁcation,” Social Networks, vol. 1,
no. 3, pp. 215–239, 1979.
[5] L.C. Freeman, S.P. Borgatti, and D.R. White, “Centrality in valued graphs: A measure of
betweenness based on network ﬂow,” Social Networks, vol. 13, no. 2, pp. 141–154, 1991.
[6] S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications, Cambridge
University Press, 1994.
[7] P.J. Carrington, J. Scott, and S. Wasserman, Models and Methods in Social Network Analysis,
Cambridge University Press, 2005.
[8] A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, and B. Bhattacharjee, “Measurement andanalysis of online social networks,” ACM SIGCOMM Conference on Internet Measurement,
pp. 29–42, 2007.
[9] M. Aida, Introduction to Network Dynamics, Morikita Publishing Co. Ltd., 2020. (in Japanese)[10] M. Aida, C. Takano, and M. Murata, “Dynamical model of ﬂaming phenomena in on-line
social networks,” IEEE/ACM International Conference on Social Networks Analysis and Mining(ASONAM 2017), pp. 1164–1171, Sydney, Australia, 2017.
[11] K. Nagatani, C. Takano, and M. Aida, “Experimental study on detecting the omen of ﬂaming
phenomena in online social networks: Theory testing of the oscillation model for online user
dynamics,” IEEE INFOCOM 2019 Workshop (CAOS ’19), Paris, France, April 2019.
[12] K. Nagatani, C. Takano, and M. Aida, “Spectral analysis of user interests for experimental
veriﬁcation of the oscillation model for OSNs,” IEEE BigData 2019, pp. 6160–6162, Los Angeles,
CA, USA, 2019.
[13] D.K. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on graphs via spectral graphtheory,” Applied and Computational Harmonic Analysis, vol. 30, no. 2, pp. 129–150, 2011.
[14] D.I. Shuman, S.K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, “The emerging ﬁeldof signal processing on graphs: Extending high-dimensional data analysis to networks and otherirregular domains,” IEEE Signal Process. Mag., vol. 30, no. 3, pp. 83–98, 2013.
[15] A. Sandryhaila and J.M.F. Moura, “Discrete signal processing on graphs,” IEEE Trans. Signal
Process., vol. 61, no. 7, pp. 1644–1656, 2013.
[16] A. Sandryhaila and J.M.F. Moura, “Big data analysis with signal processing on graphs: Representation and processing of massive data sets with irregular structure,” IEEE Signal Process.
Mag., vol. 31, no. 5, pp. 80–90, 2014.
[17] U. Von Luxburg, “A tutorial on spectral clustering,” Statistics and Computing, vol. 17, no. 4,
pp. 395–416, 2007.

531

[18] J. Bruna, W. Zaremba, A. Szlam, and Y. Lecun, “Spectral networks and locally connected
networks on graphs,” ArXiv, abs/1312.6203, 2014.
[19] D.I. Shuman, M.J. Faraji, and P. Vandergheynst, “A multiscale pyramid transform for graphsignals,” IEEE Transactions on Signal Processing, vol. 64, no. 8, pp. 2119–2134, 2016.
[20] Z. Zhang, P. Cui, and W. Zhu, “Deep Learning on Graphs: A Survey,” ArXiv, abs/1812.04202,
2018.
[21] W. Fan, Y. Ma, Q. Li, Y. He, E. Zhao, J. Tang, and D. Yin, “Graph neural networks for social
recommendation,” Proceedings of WWW ’19: The World Wide Web Conference, pp. 417–426,
2019.
[22] M.E.J. Newman, “The spread of epidemic disease on networks,” Phys. Rev., vol. E66, 016128,
2002.
[23] M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili, “Theory of rumour spreading in complex
social networks,” Physica A: Statistical Mechanics and Its Applications, vol. 374, no. 1, pp. 457–
470, 2007.
[24] J. Cannarella and J.A. Spechler, “Epidemiological modeling of online social network dynamics,”
ArXiv, vol. abs/1401.4208, 2014.
[25] R. Olfati-Saber and R.M. Murray, “Consensus problems in networks of agents with switching
topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–
1533, 2004.
[26] L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” IEEE
Transactions on Automatic Control, vol. 55, no. 4, pp. 950–955, 2010.
[27] T.A.B. Snijders, J. Koskinen, and M. Schweinberger, “Maximum likelihood estimation for social
network dynamics,” IEEE Transactions on Automatic Control, vol. 4, no. 2, pp. 567–588, 2010.
[28] M. Cha, A. Mislove, and K.P. Gummadi, “A measurement-driven analysis of information propagation in the ﬂickr social network,” Proceedings of The 18th International Conference on World
Wide Web (WWW ’09), pp. 721–730, 2009.
[29] X. Zhao, A. Sala, C. Wilson, X. Wang, S. Gaito, H. Zheng, and B.Y. Zhao, “Multi-scale
dynamics in a massive online social network,” Proceedings of The 2012 Internet Measurement
Conference (IMC ’12), pp. 171–184, 2012.
[30] R. Varga, Gershgorin and His Circles, Springer-Verlag, 2004.
[31] C. Takano and M. Aida, “Decay characteristics of user dynamics in online social networks,”
IEEE Access, vol. 8, pp. 73986–73991, April 2020.

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