Modeling Ethno-social Conflicts based on the Langevin Equation
with the Introduction of the Control Function
Alexandr Y. Petukhov, Alexey О. Мalhanov, Vladimir М. Sandalov and Yury V. Petukhov
RL “Modeling of social and political processes” Nizhniy Novgorod Lobachevski State University,
603950, Gagarin ave. 23, Nizhniy Novgorod, Russia
Keywords: Ethno-social Conflict, Society, Diffusion Equations, Langevin Equation, Communication Field.
Abstract: In this article, we propose a model of ethno-social conflict based on diffusion equations with the introduction
of the control function for such a conflict. Based on the classical concepts of ethno-social conflicts, we propose
a characteristic parameter - social distance that determines the state of society from the point of view of the
theory of conflict.A model based on the diffusion equation of Langevin is developed. The model is based on
the idea that individuals interact in society through a communicative field - h. This field is induced by every
person in a society, serves as a model of the information interaction between individuals. In addition, the
control is introduced into the system through the dissipation function. A solution of the system of equations
for a divergent diffusion type is given. Using the example of two interacting-conflicting ethnic groups of
individuals, we have identified the characteristic patterns of ethno-social conflict in the social system and
determined the effect the social distance in society has in development of similar processes with regard to the
external influence, dissipation, and random factors. We have demonstrated how the phase portrait of the
system qualitatively changes as the parameters of the control function of the ethno-social conflict change.
Using the analysis data of the resulting phase portraits, we have concluded that it is possible to control a
characteristic area of sustainability for a social system, within which it remains stable and does not become
subject to ethno-social conflicts.
1 INTRODUCTION
Ethno-social conflicts are a type of social conflict
that can be defined as a peak stage in the development
of contradictions between individuals, groups of
individuals, and society as a whole, which 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 a dialogue between two or more
parties (Dollard et al., 1993).
Ethno-social (interethnic) conflict itself can be
defined as a kind of relationship between
national/cultural groups of individuals characterized
by a confrontation in an open or latent phase (i.e. from
mutual claims to direct military or terrorist actions).
Studies on ethno-social conflicts are widely repre-
sented both in classical and modern works: (Perov,
2014; Malkov, 2004; Mason, 2013; Castellano et al.,
2009; Smith et al., 2013; Traud et al., 2011)
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 a complex interdisciplinary science. These are the
classic works of L. Coser, R. Dahrendorf, J.
Habermas, H. Becker, A. S. Akhiezer (Coser, 2000;
Darendorf, 1994), who substantiated the naturalness,
attributive character of ethno-political conflicts and
their functions in the life of society, K. Boulding, L.
Coser, P. Bourdieu (Boulding, 1969), who laid the
foundations for the construction of a general theory
of conflicts, J. Burton (Davydov, 2008) and his
followers, who turned to the problems of effective
practical technologies of the settlement and the
fundamental resolution of conflicts as a priority for
ensuring the effectiveness of conflictology, P.
Sztompka (Perov, 2014), who absolutized the
"Western, mainstream" path of social salvation, F.
Glasl, who proposed modern conflict resolution
mechanisms (Kravchenko, 2003).
330
Petukhov, A., Malhanov, A., Sandalov, V. and Petukhov, Y.
Modeling Ethno-social Conflicts based on the Langevin Equation with the Introduction of the Control Function.
DOI: 10.5220/0006853003300337
In Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pages 330-337
ISBN: 978-989-758-323-0
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

In fact, given the significant impact of such
phenomena on the society and on the processes
associated with it, the methods and ways for
describing and predicting ethno-social conflicts are
extremely important.
One of the directions for finding solutions to this
problem is the prediction and description of social
conflict by means of mathematical modeling
(Malkov, 2009; Shabrov, 1996; Blauberg and Yudin,
1973; Saati and Kerns, 1991; Bloomfield, 1997;
Plotnitskiy, 2001).
Mathematical modeling based on nonlinear
dynamics, so widely used in natural science, 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 (Abzalilov, 2012).
The models available to date can be divided into three
groups:
1) models - concepts based on the identification and
analysis of common historical patterns and their
representation in the form of cognitive schemes
that describe the logical connections between
various factors that affect historical processes
(J.Goldstein, I. Wallerstein, L.N. Gumilev, N.S.
Rozov and others). Such models generalize the
subject matter to a high degree, but they are not of
a mathematical, but of purely logical, conceptual
nature;
2) special mathematical models of imitative type,
created for the description of specific historical
events and phenomena (Yu.N. Pavlovsky, L.I.
Borodkin, D. Meadows, J.Forrester, et al.). Such
models focus on careful registration and
description of the factors and processes that affect
the phenomena under consideration. Applicability
of such models, as a rule, is limited by a rather
narrow space-time interval; they are "tied" to a
specific historical event and they cannot be
extrapolated for extended periods of time;
3) mathematical models, which are intermediate
between the two-abovementioned types. These
models describe a certain class of social processes
without claiming to provide a detailed description
of the features for each particular historical event.
Their task is to identify the basic regularities
characterizing the course of the processes of the
discussed type. In this regard, these mathematical
models are called basic models (Plotnitskiy,
2001).
Holyst J.A., Kacperski K., Schweiter F. propose a
convenient model of public opinion, which views the
interaction between individuals as a Brownian motion
(Holyst et al., 2000).
However, mathematical modeling based on
nonlinear dynamics, so widely used in natural
science, is still applied quite rarely in sociological
research.
2 PARAMETRIZATION OF
ETHNO-SOCIAL CONFLICT
It is important to identify a parameter determinant to
an ethno-social conflict, which will underlie the
model we are creating. It is clear that this parameter
should be logically justified within the framework of
the main modern concepts of social conflict.
This parameter is social distance. Previous works
(Petukhov et al., 2016) discuss this matter in more
detail; therefore, here we will only provide the
following provisions critical for understanding of this
model:
1. A major social conflict, as a rule, is accompanied
by an informational and social distance between
individuals and groups of individuals. Such a
distance can be based on interethnic, cultural,
religious, and economic differences. There can be
various reasons for such a conflict: different levels
of aggression of social and ethnic groups,
contradicting cultural and economic aspirations,
etc. Thus, the social-informational distance itself
does not cause the conflict, but, as a rule,
accompanies it.
2. This distance increases during the course of the
conflict, especially in its extreme variants
(revolutions, civil wars, etc.), leading the
opposing parties to the position of "non-
reconciliation". The history, unfortunately, has
very few examples of short and medium-term
positive scenario for such situations.
3. Therefore, this point of no return, as a rule, occurs
just before the onset of the conflict, and such a
transition of a social system from one state to
another become decisive (triggering) for the
overall situation.
In this case, as a rule, very few conflicts in a modern
globalizing world occur without external influence
and even interference. This raises the question of
introducing control into a model of conflict. This
control can play a decisive role in its generation and
dynamics.
Modeling Ethno-social Conflicts based on the Langevin Equation with the Introduction of the Control Function
331

3 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 knees
(Petukhov et al., 2016; 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 individual
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 Langevin equation of the stochastic
diffusion, which has been applied with relative
success for modeling of some social processes as
well. For example, the previously mentioned model
(Holyst, Kasperski, 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 for 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 et al.,
2000), but with another parametrization and another
type of initial equations). This field is induced by
each individual in society and serves as a model of the
information interaction between individuals.
However, we should keep in mind that here we are
talking about a society, which is difficult to classify
as 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 caused by the fact that in the modern information
world there is no need to be close to the object of
influence in order 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 into the information exchange. The
proximity of individuals to each other in this model
suggests that there is a regular exchange of
information between them, which establishes a social
connection. 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 (Perov, 2014), 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 a number of physical analogies that are
similarly influenced by physical systems, for
example, a dissipative function that can have different
forms in different physical conditions (Malkov,
2009).
4 MATHEMATICAL
REPRESENTATION OF THE
SYSTEM
The communicative field, as in (Petukhov et al.,
2016), is represented by a diffusion equation with a
divergent type of diffusion:


ℎ
(


,
)
=

,



,


̅
(





),(





)


+ℎ
(


,
)
−ℎ
(


,

)
,
(1)
where 

,

 is a function that describes the
interaction between individuals, which is modeled by
SIMULTECH 2018 - 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
332

the classical Gaussian distribution;
2
2
()
1
(, )
ij
xx
ij
xx e
ε
ϑ
επ
−−
=
,
Function 

,

 is introduced instead of the
delta-function to simplify the process of computer
modeling;

̅
(





),(





)
is the inverse Kronecker symbol;
 is the diffusion coefficient describing the
propagation of the communicative field.
The movement of an individual in space is described
by the Langevin equation:



=(

)+









ℎ

,

,

+
√
2

(

)
,
(2)

(

)
is the control function, which we set as:

(

)
=−



where  is the time of relaxation in the society,

с

- coefficient of social activity of the ith
individual or a group of individuals,



– coefficient of the scientific and technological
progress of the th individual or a group of
individuals,


(

)
−stochastic force.
We believe that the distinctive parameters of the
system can take on values:
0<

,

,<1.
In the general case, the following are chosen as
the initial conditions for equations (1) and (2):


|

=

,
ℎ
(


,=0
)
=ℎ

.
5 APPROXIMATE SOLUTION OF
THE SYSTEM
Let us consider a model of two interacting
consolidated ethnic groups of individuals,
presumably in a state of conflict. In this case,
equations (1) and (2) produce four equations that fully
describe the model of interaction of individuals:
()
()( )
[]
()
()
()( )
[]
()
()
()
()
()
2
2
12
2
2
2
12
2
1
21
1
11
1
12
2
22
11
2
1
11
2
22
1
2
22
1
()
,
,,0 ,
,
,,0
,
2
2,()
,
,
xx
cs
xx
cs
cs
cs
hxt
Dhx t hx kke
t
hx t
Dhx t hx kke
t
hx t
dx
xkk Dt
dt x
hxt
dx
xkk Dt
u
dt x
u
ψ
ψ
ψ
ψ
α
α
ξ
ξ
+
−−
+
−−
∂
=−+
∂
∂
=−+
∂
∂
=+
∂
∂
=++
∂
+















(3)
where:
.
1
,
2211
,
2211
scsc
kkkk
scsc
kkkk
++
=+++= δ
πψ
α
ψ
Here, as in (Petukhov et al., 2016): in order to obtain
approximate analytic solutions of the system (3), we
use the series expansion accurate to first-order
quantities of smallness for
ioi
x
xxΔ= −
o
tttΔ= −
difference:
0
0
0,
0,
(,) ( ,) ( ) ( )
ii
ii
ioio
txx
i
txx
hh
hx t hx t x t
xt
==
==
∂∂
−≈ Δ+ Δ
∂∂
,
Then, assuming that the following initial conditions
are present:
0
0
0,
0,
0, ( , ) ( ) ( ) 1
ii
ii
oi oi o
txx
i
txx
hh
xhxt
xt
==
==
∂∂
== = =
∂∂
,
let us integrate the first two equations of the system
(3), and then, using the obtained results and the two
latter equations of the system (3), considering the
continuity of the corresponding functions, transform
the system. Let us then differentiate over time.
Assuming that the stochastic forces for the two
groups are the same
() ()
tt
21
ξ
ξ
=
.
Then, by introducing new variables:
()
()
()
,
1
,
1
2
,
,
2
2
12222111
2
2
2211
21
ψ
ψ
ψ
ψ
α
+
=
+
+
=
−=
−=
C
kkkkkkkkB
kkkkDA
xxy
scscscsc
scsc
we obtain an equation that looks as follows:
Modeling Ethno-social Conflicts based on the Langevin Equation with the Introduction of the Control Function
333






=

−


+


,>0,>0,
=
1

,
(4)
where ,, depend on the parameters: 


,


,.
Let us write the equation (4) in the Cauchy form:



=,


=

−+


.
(5)
The system (5) can be viewed as a dynamic system
that describes the process of interaction of two
individuals or groups of individuals. This system is
non-conservative, but finding its equilibrium states is
reduced to solving the same system of equations as in
the conservative case, see (Petukhov et al., 2016):

=0,



=−


.
(6)
It was shown in (Petukhov et al., 2016) that the
corresponding system has two equilibrium states: the
saddle and the center. The general theory of
dynamical systems states that the saddle is a rough
equilibrium state, that is, its type does not change
after a sufficiently small change in the system. While
the center is a non-rough state of equilibrium, with
small changes in the system, such a state of
equilibrium shifts to a stable or unstable focus.
Taking into account the discussion of rough and
non-rough equilibrium states, it is easy to construct a
phase portrait of the system under consideration in the
presence of dissipation (Figure 1. Considering the
above, the equilibrium state 

of the saddle type
does not change its type, but the stable separatrix loop
will break, while the equilibrium state 

of the center
type >0 (>0) will shift into a steady focus.
Figures 1-6 show phase portraits for the case of
two equilibrium states under conditions
0<−


<

1
2




,

<0
(7)
or conditions
−

1
2




<−


<0,

>0
(8)
for three different values of the parameter  (


;1;2),
where =


.
Figure 1: Phase trajectories under conditions (7) and =


.
Figure 2: Phase trajectories under conditions (8) and =


.
The obtained phase portraits show that 

is the
stable node and 

is the saddle. Separatrix, which
passes along the border between the gray and white
parts of the Figure, refers to the saddle 

. These
separatrices divide the phase plane into areas with
qualitatively different behavior of the phase
trajectories. The area highlighted in gray is the area
of asymptotic stability of the node (region of
attraction).
Figure 3. Phase trajectories under conditions (7) and =1.
SIMULTECH 2018 - 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
334

Figure 4: Phase trajectories under conditions (8) and =1.
Figure 5: Phase trajectories under conditions (7) and =2.
Figure 6: Phase trajectories under conditions (8) and =2.
The obtained phase portraits (Figure 3 – Figure 6)
show that 

is the steady focus and 

is the saddle.
Trajectories between the gray and white "zone" of the
Figure are the separatrices of the saddle 

. The gray
area is the region of asymptotic stability of the node
(region of attraction).
The analysis of the obtained results (see Figure 1,
Figure 6) can lead to the conclusion that there is a
certain region of asymptotic stability. Figure 4-6
shows that with increasing of the parameter  (which,
from the physical point of view, corresponds to a
decrease in the impact of the dissipation function), the
region of attraction of the stable equilibrium state
decreases, which agrees with the classical concept of
the coefficient of friction. The edges of this region are
the separatrices of the saddle 

.
When changing the parameters, we can easily
notice that the behavior of the phase trajectories also
changes. This concerns purely quantitative changes in
the size and location of trajectories, but can lead to
significant, qualitative changes in the structure of the
phase portrait, i.e. bifurcation. For example, under the
following conditions:
0<−


<

1
2




,<0,
we have
(
Figure1 − 6
)
two simple singular points


and 

, but when we reach the value:
−


=







,<0,
equilibrium states 

and 

merge, forming one
complex singular point, which, if the following
conditions are met:
−


>

1
2




,<0,
does not appear at all. Thus, bifurcation here is
characterized by the birth and disappearance of
equilibrium positions. In the model under
consideration, the bifurcation values of the
parameters are as follows:
−


=0,−


=

1
2




,
−


=−







.
Individuals or groups of individuals, who have the
necessary parameters to enter the area of asymptotic
stability at the initial moment of time remain at a
distance, within which social connections and active
information exchange are possible, which means that
a conflict state is unlikely or impossible.
As noted in the statement of the task, in a
society, where social and informational contact, as
well as the interpenetration of different cultures and
ethnic groups are sufficient, where separate groups of
people do not separate from each other creating
closed subsystems (where the conditions differ
Modeling Ethno-social Conflicts based on the Langevin Equation with the Introduction of the Control Function
335

significantly from the basic system), the possibility of
the emergence of ethno-social, religious and other
conflicts is reduced to a relative minimum.
Individuals or groups of individuals that have
fallen outside the region of stability at the initial
moment, over time, will end up at a relatively large
social distance. This particular state of the social
system can be described as the conflict and the
manifestation of the existing contradictions between
individuals and groups of individuals (Petukhov et al.,
2016). For example, in ethno-social conflicts, this is
manifested in the minimization of social and cultural
contacts between different ethnic groups, the increase
in the socioeconomic gap, growing contradictions
and, as a result, the transition to an open confrontation
phase with the destabilization of the social and
political system as a whole.
The control function for an ethno-social conflict
u(x) (See (2)) introduced here demonstrates how, with
a change in its parameters, the phase portrait, and
therefore the state of the social system can be
substantially changed. This suggests that with a
certain mediation, it is possible to achieve a "larger"
stability zone, which will attract a greater number of
phase trajectories, which in turn provides a greater
chance of maintaining the necessary social distance in
order to minimize the chances of an ethno-social
conflict.
6 CONCLUSIONS
Social hyper-clusterization of society, sharp division
in the information and social environment of the
coexistence of individuals, and cultural and
interethnic dissociation create ideal conditions for
social conflict. The prevention of conflicts in society,
the definition of their triggers and the search for the
most effective scenarios for their suppression are the
important tasks for modern social sciences.
This article briefly reviewed the main approaches
to modeling in the social sciences, the problems of
determining social conflict and its main concepts. A
formalized definition of one of the parameters leading
to a conflict in the social system is given.
A mathematical model based on the Langevin
equation is proposed, an analytical solution is given
in the first approximation for a divergent diffusion
type. The function of management (mediation) by
conflict is introduced based on the physical analogy -
the dissipation function.
Specific trigger conditions that take into
consideration the external influence and control were
established. These conditions are determined by the
parameters of the social system, under which the
grounds for the emergence of social conflict and its
aggravation are created.
Modeling of the system allowed identifying a
distinctive region of stability for the social system,
determined by phase trajectories. In this area, the
studied objects maintain a relatively short social
distance between each other, which is typical for
social groups, which are actively interacting and stay
in a constant information contact. It has been shown
how, depending on the impact of the conflict control
function, this region is changing.
By determining and correlating these trigger
states with the introduced parametrization of the
control function, it is possible to determine the
patterns corresponding to certain modern ethno-
social conflicts, which makes it possible to use this
model as a tool for predicting their dynamics and the
formation of resolution scenarios.
ACKNOWLEDGEMENTS
The study was supported by grants from the Russian
Foundation for Basic Research No. 17-06-00640_a
and 16-29-09550_ofi_m.
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