Modelling Social Protests in the Republic of Belarus in 2020 based on
Diffusion Equations
Alexandr Y. Petukhov
1a
and Dmitry I. Kaminchenko
2b
1
Keldysh Institute of Applied Mathematics Russian Academy of Science, Miusskaya sq., 4, Moscow, 125047, Russia
2
RL “Modeling of Social and Political Processes” Nizhniy Novgorod Lobachevski State University,
603950, Gagarin ave. 23, Nizhniy Novgorod, Russia
Keywords: Protests, Belarus, Society, Diffusion Equations, Langevin Equation, Communicative Field, Social Activity,
Networks.
Abstract: In this article, we propose a model of social activity based on diffusion equations and a comparison of the
modeling results with real data of protest activity (based on data in social networks) in Belarus in 2020. A
model uses the diffusion Langevin equation. The model is based on the idea that individuals interact in society
through a communicative field - h. Besides, the control is introduced into the system through the dissipation
function. Protest data indicators were collected using the authors' content analysis of the main hashtags
associated with Belarusian protests. Then the results of modeling were compared with the obtained data and
analyzed. Based on the modeling we have revealed a general similarity in dynamics and characteristic patterns
as well as have made a forecast for the development of the situation in 2021.
1 INTRODUCTION
Social conflict can be defined as a peak stage in the
development of contradictions between individuals,
groups of individuals, and society as a whole, is
characterized by the existence of conflicting interests,
goals, and views of the subjects of interaction.
Conflicts may be hidden or explicit, but they are
always based on the absence of compromise, and
sometimes even arise from a dialog between two or
more parties (Dollard et al. 1993). The development
of general conflictology at the present stage was
significantly influenced by the works of international
scientists, who had laid the theoretical foundation for
solving specific problems of complex
interdisciplinary science. These are the classic works
of L. Coser, R. Dahrendorf, J. Habermas, H. Becker,
A. S. Akhiezer as well as other studies on social
conflicts (Dahrendorf, 1965; Gurr and Harff, 1994;
Galtung, 1969; Gurr, 1993; Greenfeld, 1992; Isajiw,
1974; Boulding, К, 1969; Krisberg, 1998), modeling
of social processes (Castellano, Fortunato, and
Loreto, 2009; Smith, 2003; Traud, 2011).
a
https://orcid.org/0000-0002-7412-5397
b
https://orcid.org/0000-0002-3193-3423
Mathematical modeling based on nonlinear
dynamics is widely used in natural science, but it is
still applied quite rarely in sociological research. In
recent years, significant progress has been made in
the development of models of social and political
processes (Plotnitskiy, 2001).
As a rule, the modeling of the dynamics of the
linear system in classical models is based on the use
of multidimensional equations, difference equations,
the mathematical apparatus of cellular automata, the
mathematical apparatus of catastrophe theory, the
mathematical apparatus of self-organized criticality
theory, the stochastic differential Langevin equations
and Itô-Stratonovich, the analysis of systems with
chaos and reconstruction of stable states (attractors)
by time series (Malkov, 2009; Romanovsky
Stepanova and Chernavsky, 1984; Haken, 1985,
Malinetskiy and Potapov, 2000).
A certain class of works was devoted to ethnic
diversity and its impact on economic and
sociocultural development, as well as other social
processes associated with it (Shabrov, 1996;
Ottaviano and Peri, 2005; Weber, Davydov and
Dower, 2015). These are the interdisciplinary
Petukhov, A. and Kaminchenko, D.
Modelling Social Protests in the Republic of Belarus in 2020 based on Diffusion Equations.
DOI: 10.5220/0010612104450452
In Proceedings of the 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2021), pages 445-452
ISBN: 978-989-758-528-9
Copyright
c
2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
445
researches dealing with social problems and their
correlation with the economy taking into account
ethno-cultural factors, as well as their joint influence
on potential and existing conflicts.
Forecasting and description of socio-political
processes are carried out by many other methods
(Mikhailov and Gorbatikov, 2012; Smith, 2003;
Traud, 2011).
In fact, given the significant impact of such
phenomena on the society and the processes
associated with it, the methods and ways for
describing and predicting ethno-social conflicts are
extremely important.
In recent years, significant progress has been
made in the development of models of social and
political processes (Abzalilov, 2012). Mathematical
modeling based on nonlinear dynamics is widely used
in natural science, but it is still applied quite rarely in
sociological research.
2 FUNDAMENTALS OF THE
MODEL
Socio-political processes are subject to constant
changes and deformations, therefore from the point
of view of mathematical modeling they cannot be set
with a high degree of precision. Here we can trace
the analogy with the Brownian particle, i.e. a
particle that seemingly moves along a rather defined
trajectory, but under close examination, this
trajectory turns out to be strongly tortuous, with
many small fractures (Holyst, Kasperski and
Schweitger, 2000; Petukhov et al. 2018; Gutz and
Коrobitsyn, 2000). These small changes
(fluctuations) are explained by the chaotic motion of
other molecules. In social processes, fluctuations
can be interpreted as manifestations of the free will
of its participants, as well as other random
manifestations of the external environment (Gutz
and Коrobitsyn, 2000).
In physics, these processes are, as a rule,
described by the Langevin equation of the stochastic
diffusion, which has been relatively approved for
modeling of some social processes as well. For
example, the model of public opinion, developed by
Holyst J.A., Kacperski K., Schweiter F. (Holyst,
Kasperski and Schweitger, 2000), is based on the use
of this equation.
This approach has several advantages:
1. As it has already been mentioned, the
approach allows taking into account the
manifestations of the free will of its individual
participants, as well as other random
manifestations of the external environment for
the social system.
2. The behavior of a social system can be
calculated, both for its entirety and separate
individuals.
3. This approach allows identifying some
distinctive stable modes of functioning of
social systems, depending on various initial
conditions.
4. Diffusion equations, as a mathematical
apparatus, have been sufficiently validated
and studied from the point of view of
numerical simulation.
The model is based on the assumption that
individuals interact in society through a
communicative field - h (a similar concept was
introduced in (Holyst, Kasperski and Schweitger,
2000), but with another parametrization and another
type of initial equations).
The need to introduce this concept was caused by
a number of factors:
1. Any alive cognitive system (including both
society and a person) carries out the activity
and functioning on the basis of the exchange
of information in one form or another. It can
be either information from electrical impulses
in the brain or the cerebellum opioid system,
or information flows on the Internet.
2. Therefore, modeling of individuals’ behavior
in society will be directly related to modeling
their information exchange.
3. Accordingly, it is reasonable to introduce a
function as the basis of the model that will
simulate the information exchange between
individuals.
This is the reason why the function h, in fact, the
function of information exchange between
individuals, was developed. Physically, it is a field
created by every person in society, simulating
information interaction. Besides, from the point of
view of physical and mathematical description of this
field, we should keep in mind that here we are talking
about a society, which is difficult to refer to an object
in classical physical spatial topology. Objectively,
from the point of view of information transfer from
an individual to an individual, space in society
combines both classical spatial coordinates and
additional specific parameters and features. This is
associated with the fact that in the modern
SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
446
information world there is no need to be close to the
object of influence to transmit information to it.
Thus, the society is a multidimensional, social-
physical space that reflects the ability of one
individual to "reach" another individual with his
communicative field, that is, to influence it, its
parameters, and the ability to move in a given space.
Accordingly, the position of the individual relative to
other individuals in such a space, among other things,
models the level of relationships between them and
involvement in the information exchange. The
proximity of individuals to each other in this model
suggests that there is a regular exchange of
information between them and a social connection
has arisen. The conflict in such a statement of the
problem should be regarded as a variant of the
interaction of individuals, or groups of individuals, as
a result of which the distance (i.e., social distance xi -
xj, where xi and xj are the coordinates in social and
physical space, i, j = (1, N), where N is the number of
individuals or consolidated groups of individuals)
between them is growing rapidly.
Conflict management or various options for
conflict mediation (Dahrendorf, 1965; Gurr and
Harff, 1994; Isajiw, 1974; Boulding, К, 1969), from
the point of view of modeling, are an additional
function that depends at least on the coordinates and
affects the overall stability and structure of the social
system. There are several physical analogies that
similarly influence physical systems; for example, a
dissipative function that can have different forms in
different physical conditions (Holyst, Kasperski and
Schweitger, 2000).
2.1 Mathematical Representation of
the System
The communicative field, as in (Petukhov et al.
2018), is represented by a diffusion equation with a
divergent type of diffusion:
𝜕
𝜕𝑡
ℎ
(
𝑥
,𝑡
)
=𝑓𝑥
,𝑥
𝜗𝑥
,𝑥
𝛿
̅
(
),(
)
+𝐷ℎ
(
𝑥
,𝑡
)
−ℎ
(
𝑥
,𝑡
)
,
(1)
where f(x_i,x_j ) is a function that describes the
interaction between individuals, which is modeled by
the classical Gaussian distribution:
This interaction is an information exchange
between individuals, which is carried out through any
communicative physical means.
𝜗(𝑥
,𝑥
)=
√
𝑒
(
)
,
Function 𝜗𝑥
,𝑥
is introduced instead of the
delta-function to simplify the process of computer
modeling;
𝛿
̅
(
),(
)
– is the inverse Kronecker symbol;
𝐷 – is the diffusion coefficient describing the
spread of the communicative field.
The movement of an individual in space is
described by the Langevin equation:
𝑑𝑥
𝑑𝑡
=𝑢(𝑥
)+𝑘
𝑘
𝜕
𝜕𝑥
ℎ𝑥
,𝑡
,
+
√
2𝐷𝜉
(
𝑡
)
,
(2)
𝑢
(
𝑥
)
– is the control function, which we set as:
𝑢
(
𝑥
)
=−
𝑥
𝜏
where 𝜏− is the time of relaxation in the society,
𝑘
с
– coefficient of social activity of the i
th
individual or a group of individuals,
𝑘
– coefficient of the scientific and technological
progress of the i
th
individual or a group of individuals,
𝜉
(
𝑡
)
− stochastic force.
We believe that the distinctive parameters of the
system can take on values:
0<𝑘
,𝑘
,𝐷<1.
In the general case, the following are chosen as
the initial conditions for equations (1) and (2):
𝑥
|
=𝑥
,
ℎ
(
𝑥
,𝑡=0
)
=ℎ
.
The field ℎ
(𝑥,𝑡) affects an individual I in the
following way. Being at the point x
i
, the individual
falls under the influence of the communication field
of another individual (or several). Depending on the
difference between its coefficients and the
coefficients affecting individuals, it can react in the
following ways:
1. Changes the value of its coefficients under the
influence of other individuals
2. Moves in the direction of the area where the
difference of the coefficients is relatively
minimal at the moment
Modelling Social Protests in the Republic of Belarus in 2020 based on Diffusion Equations
447
Let us consider 𝑝
(𝑘
,𝑘
,𝑡,𝑥
,𝑥
) as the
probability of the impact of the communication field
of an individual (or a cluster of individuals) j on the
communication field of an individual i in a way to
change its coefficients K
s
and K
c
(separately or
together) at time t. In this case, the probability of
movement of an individual i in the direction of the
area where the difference of the coefficients is
relatively minimal at present -1−𝑝
(𝑘
,𝑘
,𝑡,𝑥
,𝑥
).
The change in probability then:
𝑑
𝑑𝑡
𝑝
(𝑘
,𝑘
,𝑡,𝑥
,𝑥
)
=𝜐(𝑘
|
𝑘
′
)
′
𝑝
(𝑘
′
,𝑘
′
,𝑡,𝑥
,𝑥
)𝜗(𝛥𝑥
𝛥𝑘
)−
−𝑝
(𝑘
,𝑘
,𝑡,𝑥
,𝑥
)
∑
𝜐(𝑘
′
𝑘
)
′
𝜗(𝛥𝑥
,𝛥𝑘
′
),
(3)
𝜗(𝛥𝑥
𝛥𝑘
) – a parameter characterizing the
induction effect of the communication field.
𝜐(𝑘
|
𝑘
′
) – abstract probabilities of changing the
coefficients per unit of time:
𝜐(𝑘
′
𝑘
)=
𝑘
≠𝑘
′
→𝜂𝑒𝑥𝑝ℎ
′
(𝑥
,𝑡)−ℎ
(𝑥
,𝑡)/𝑄
𝑘
=𝑘
′
→0
,
(4)
где Q – a parameter of social freedom
characterizing the degree of freedom of movement of
individuals in a given social system.
The next stage of the study must be focused on the
data of protests in Belarus, for the reason that the
results of the modeling are of special interest for us in
this study, and then compare them with real data.
3 SOCIAL MEDIA AS A
SUBSYSTEM OF THE MODERN
POLITICAL AND
COMMUNICATIVE SPACE
If we consider the general field of modern political
communication as a separate system, so the political
and communicative space of social media is its
subsystem. Such features of communicative
interactions as dynamism, interactivity, and
connectivity have a special manifestation in this
subsystem. This subsystem is closely connected with
the other subsystems of political communication
which often play the role of an external environment
to it, sending its impulses that cause a certain reaction
within this subsystem. The example of a similar
reaction in the political and communicative space of
social media can be noticed in the course of the
protest events in the Republic of Belarus.
The choice of this social conflict is caused by its
relevance, sufficiency of informational
representation, and active presentation on social
media. The experience of previous social conflicts
with high integration of social media demonstrates
that the activity of participants in the process on the
internet can serve as a marker for determining their
level of involvement (Erz, Marder and Osadchaya,
2018; Lidgren, 2019; Bonilla and Rosa, 2015).
To conduct our research, we carried out a
quantitative content analysis of the subject of posts
published by residents of the Republic of Belarus in
the support system of modern social networks
Facebook, dedicated to protesting actions that began
immediately after summing up the results of the
presidential elections in Belarus on 9 Aug. 2020. We
have chosen a message (Facebook users’ notes) as a
unit of analysis and a word as a unit of calculating
(keyword, hashtag).
When writing texts of messages on social media,
several specific hashtags are often used (“a keyword
or phrase indicated by a hash sign that turns this
word/phrase into a hyperlink” and reflects the subject
and content of the message (Erz, Marder and
Osadchaya, 2018, p. 50). During the reflection of the
protest events in Belarus in the Facebook information
field, a number of certain hashtags became integral
elements of the symbolic semantic core of the
communication activity of the republic's residents on
Facebook. Among the designated hashtags are the
following: “#Belarus2020”, “#Беларусь2020”,
“#ЖывеБеларусь” (“#Long live Belarus”),
“#ВерымМожамПераможам” (“#believecanwin”).
As it is known, hashtags are not just words or
expressions corresponding to the used sign, but,
“tools of activating certain interpretive frames”,
according to S. Lindgren (Lidgren, 2019, p. 421).
They play a semiotic role pointing out the intended
meaning of the utterance, allowing users to state in
the message that semantic meaning, which,
otherwise, might not be so obvious (Bonilla, Y., and
J. Rosa, 2015, p.5). Thus, in some cases, hashtags
mark the corresponding user posts with a certain
value, fixing a meaningful message to the audience in
the post itself, or with a certain meaning, stating a
meaningful message to the audience in the post.
The time interval for content analysis is from 11
Aug. to 2 Oct. During the indicated period, 604
messages were selected including the hashtags
SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
448
mentioned above as a component element. It is
necessary to emphasize that we have analyzed only
messages published by residents of the Republic of
Belarus based on open data on Facebook.
3.1 Results of Content Analysis
The dynamics of changes in the frequency of posting
messages with the indicated hashtags in general by
days is presented in Figure 1.
The dynamics of changes in the frequency of
posting messages with the indicated hashtags
separately by days is presented in Figure 2.
Figure 1.
In messages dedicated to protest activity in
Belarus, it was more often used (in comparison with
others) «#ВерымМожамПераможам» (213
messages) and «#Belarus2020» (202 messages) by
residents of the Republic. The hashtag
«#ЖывеБеларусь» was used in 164 in posts and
«#Беларусь2020 – 25 in messages.
Figure 2.
4 RESULTS OF THE MODELING
The modeling was carried out in the MatLab 2013b
environment. Two variants with different initial
conditions were served as a basis for research.
The first variant (Figure 3) represents the
modeling of social indignation without any external
influence (without the control function), in order to
Along axes: axis y – change in social activity Δkc, axis x –
timing t.
Figure 3.
0
5
10
15
20
25
30
35
40
11 August
15 August
19 August
23 August
27 August
31 August
4 September
8 September
12 September
16 September
20 September
24 September
28 September
2 October
Number of messages
Number of messages with hashtags
#Belarus2020, #ЖывеБеларусь,
#Беларусь2020,
#ВерымМожамПераможам в
Facebook
0
2
4
6
8
10
12
14
16
18
20
11 August
17 August
23 August
29 August
4 September
10 September
16 September
22 September
28 September
Number of messages
Number of messages with hashtags
#Belarus2020, #ЖывеБеларусь,
#Беларусь2020,
#ВерымМожамПераможам в
Facebook
#Belarus20
20
#ЖывеБел
арусь
#Беларусь
2020
#ВерымМо
жамПерам
ожам
Modelling Social Protests in the Republic of Belarus in 2020 based on Diffusion Equations
449
analyze the variant of a closed system and to make a
conclusion on how the internal social conflict should
proceed in this case. The modeling results are similar
to earlier works on the dynamics of social activity in
conflict conditions (Petukhov 2020).
The second situation refers to a variant of social
disturbance with external influence/control creating a
sequential series of “disturbances” in the
communication field of the social system (Figure 4).
Along axes: axis y – change in social activity Δkc, axis x –
timing t.
Figure 4.
4.1 Data Analysis and Comparison
We have analyzed the intensity of use of the
designated hashtags on Facebook (for the period from
11 Aug. to 2 Oct.) and identified a number of days
when some of these hashtags were used especially
intensively: 9, 20, 21, 23, and 27 September. Only 20
and 27 Sept. were days off, which is important to
emphasize because the main rally activity took place
in Belarus exactly on weekends. Therefore, the active
use of the hashtags mentioned above could become a
reaction to events that happened in the external
environment (towards the political and
communicative field of Belarus).
The reason for the active use of thematic hashtags
on 9 Sept. could become a speech of one of the leaders
1
Nikolaev P. Sanctions against the authorities:
Tsikhanouskaya spoke at the PACE // Internet edition
Gazeta.ru. URL: https://www.gazeta.ru/politics/2020/09/
08_a_13241324.shtml
2
Krayushkins M United Nations Human Rights Council
condemned what is happening in Belarus // Internet
edition Gazeta.ru. URL: https://www.gazeta.ru/social/
news/2020/09/18/n_14958775.shtml
3
Kazantseva К. The US does not officially recognize
Lukashenko as the legitimate president of the Republic of
of the Belarusian opposition Svetlana
Tsikhanouskaya at the Parliamentary Assembly of the
Council of Europe (PACE) on 8 Sept. During her
speech, Svetlana Tsikhanouskaya made an appeal to
PACE for imposing sanctions against the leadership
of Belarus
1
.
An important external factor that contributed to
the increase in communication activity on Facebook
(using the considered hashtags) in the period from 20
to 23 Sept. could be the resolution of the United
Nations Human Rights Council, adopted at the
meeting held on 18 Sept. The resolution condemns
violations of human rights in Belarus, calling on the
country's authorities to take measures to resolve the
situation
2
. A statement by a representative of the US
State Department was released directly on 23 Sept.
According to this statement, the US did not officially
recognize A.G. Lukashenko as the legitimate
president of the Republic of Belarus
3
. On the same
day the official representative of the German
government S. Seibert, the head of the Czech
Ministry of Foreign Affairs T. Petršicek, and the head
of the Danish Foreign Ministry J. Kofod also declared
that their states did not recognize the legality of A.G.
Lukashenko for the presidency of the Republic of
Belarus
4
.
On 27 Sept. French President E. Macron declared
in an interview with a weekly Journal du Dimanche
that, in his opinion, the Belarusian leader A.G.
Lukashenka must resign
5
. It provoked a response
from the leadership of Belarus and can also be viewed
as an informational influence of the external
environment on the political and communicative
space of Belarus.
When comparing curve trajectory in Figure 1 and
Figure 2 with Figure 3 and Figure 4, it is necessary to
emphasize the repeated cyclicity in Figure 1, Figure 2
and cyclicity of the same nature in Fig. which, as
noted, demonstrates the reaction of the
communicative field (subsystem) to external
information impulses. They cause an increase in the
intensity of social activity and contribute to the
activation of communicative actions of social media
users with the use of certain content-semantic
Belarus // Internet edition Gazeta.ru. URL: https://www.
gazeta.ru/politics/news/2020/09/23/n_14981911.shtml
4
Fakhrutdinov P. "What a farce": Europe condemned
Lukashenka's inauguration // Интернет- Internet edition
Gazeta.ru. URL: https://www.gazeta.ru/politics/2020/09/
23_a_13263973.shtml
5
Ermolov A. "Lukashenko must leave": Macron accused
Minsk of authoritarianism // Internet edition Gazeta.ru.
URL: https://www.gazeta.ru/politics/2020/09/27_a_132
70117.shtml
SIMULTECH 2021 - 11th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
450
(thematic) verbal tools (hashtags).
Also, a comparison of the results and general
characteristic patterns indicates the similarity of real
data (from the point of view of repetition of growing
peaks of activity) with the simulation results in Figure
4, which from a perspective of the presented concept
of the approach implies an increasing cyclical
external interference in the social conflict in Belarus.
Based on the results of modeling, we should
predict (at the time of the survey in Nov. 2020) that
the main peak of social disturbance is likely still
ahead. Of course, various additional factors, such as
the coronavirus epidemic, can interfere in the process.
However, from the point of view of the model, the
process has not yet reached the main maximum and
its appearance can be assumed at the beginning of
2021 if external influence and main internal trends
persist.
5 CONCLUSIONS
This article proposed a model based on diffusion
equations (Langevin equations, in particular) that
models information and communication interactions
of individuals in society in various conditions. One of
the main parameters of the model is the coefficient of
social activity, and the change of this parameter can
be indicative in the context of ongoing social conflicts
from the point of view of analyzing the state of
society.
In order to test the model, we have chosen a
specific real situation - public protests in Belarus in
2020.
It also should be pointed out that informational
influences from the external environment can make
an impact on the political and communicative field of
the political system, causing a response from the
communication participant. In particular, one of the
consequences of such an impact is an increase of user
activity in the context of modern social media support
platforms, which use special symbolic elements,
certain hashtags, in the process of text
communication acts. According to the conducted
analysis of the communicative actions of the
inhabitants of Belarus on Facebook certain events (of
a certain nature and political orientation) that took
place in the information field of foreign states and
international organizations, often preceded (and/or
accompanied) the active use of the appropriate
hashtags.
The results of the conducted content analysis
partially correspond to the results of modeling,
especially the general pattern in Figure 4, which
makes it possible to assume that there is an external
influence/interference in conflict processes in the
Republic of Belarus.
We have also made a forecast for the development
of the situation and the possibility of a larger
disturbance in the social system when/while
maintaining the current trends and the level of
external influence.
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