Mathematical Modeling of the Ethno-social Conflicts by Non-linear
Dynamics
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: The issue of modeling various kinds of social conflicts (including ethno-social) using diffusion equations is
discussed. The main approaches to and methods of mathematical modeling in contemporary humanitarian
sciences. The main concepts of social conflicts, ways of their classification, interpretation, including ethnic-
social, religious and other conflicts are considered. The notion of a conflict in a social system is defined in
terms of mathematical modeling. A model based on Langevin diffusion equation is introduced. The model is
based on the idea that all individuals in a society interact by means of a communication field - h. This field is
induced by each individual in the society, modeling informational interaction between individuals. An
analytical solution of the system of thus obtained equations in the first approximation for a diverging type of
diffusion is given. It is shown that even analyzing a simple example of the interaction of two groups of
individuals the developed model makes it possible to discover characteristic laws of a conflict in a social
system, to determine the effect of social distance in a society on the conditions of generation of such processes,
accounting for external effects or a random factor. Based on the analysis of the phase portraits obtained by
modeling, it is concluded that there exists a stability region within which the social system is stable and non-
conflictive.
1 INTRODUCTION
A social conflict can be defined as a peak stage in the
development of contradictions in relations between
individuals, groups of individuals, or a society as a
whole, characterized by the presence of contradicting
interests, objectives and viewpoints of the interacting
subjects. Conflicts can be latent or explicit, and are
caused by lack of a compromise or sometimes even a
dialogue between the two or more parties involved
(Petukhov, 2015a).
The English sociologist E. Giddens introduced the
following definition of a conflict: “a social conflict is
understood as real struggle between interacting
people or groups, no matter what its causes, ways and
means used by each of the involved parties are”.
Works of the foreign scientists that became
fundamental in analyzing practical problems of this
complex inter-disciplinary science played an
important part in the development of general
conflictology at the present stage. These are the
classical works of L. Coser, R. Darendorf, U.
Habermas, G. Bekker, A.S. Ahiezer, who
substantiated a natural and attributive character of
ethnic-political conflicts and their functions in the life
of a society; K Boulding, L. Kozer, P. Bourdier, who
laid the groundwork for developing a general theory
of conflicts; J. Burton and his followers, who
addressed the ussie of effective practical technologies
of settling and principal resolution of conflicts as the
first-priority one for making the conflictological
knowledge effective; P. Schtompke, who absolutized
the “western main road” of social salvation; F. Glazl,
who introduced modern mechanisms of solving
conflicts.
The issues of studying, classifying and, most
important, predicting conflicts have always been of
importance in fundamental sociology. This issue was
addressed in numerous works of the leading
sociologists and mathematicians: J. Bernard, R.
Bailey, K. Boulding, D. Bucher, J. Duke, L. Coser, L.
Krisberg, D. Leidis, R. Makk, A. Rapoport, R.
Snamayer, R. Stagner, T. Shelling, T. Bottmore, J.
Rex, G. Boutoul, M. Crosieau, A. Touren, K.
Darendorf, E. Vyatr, Y. Moukha, Y. Sctumski, Y.
Reykovski, L.A. Nechiporenko, I.I. et al. (Davydov,
2008; Kravchenko, 2003; Shabrov, 1996).
180
Petukhov, A., Malhanov, A., Sandalov, V. and Petukhov, Y.
Mathematical Modeling of the Ethno-social Conflicts by Non-linear Dynamics.
DOI: 10.5220/0006393501800187
In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 180-187
ISBN: 978-989-758-265-3
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
Taking into account the important impact of such
phenomena on a society and all the processes inside
it, any ways of predicting and discovering
characteristic laws of social conflicts are certainly of
a paramount importance.
One of the directions of searching for possible
solutions of this problem is forecasting and describing
a social conflict mathematically, i.e., using
mathematical modeling (Shabrov, 1996; Mason,
2013; Blauberg, 1973; Saati, 1991; Bloomfield, 1997;
Plotnitskiy, 2001).
2 MATHEMATICAL MODELING
IN SOCIAL SCIENCE
Mathematical modeling based on nonlinear
dynamics, which is widely used in natural sciences, is
still rather rarely resorted to in sociological studies.
In the recent years, considerable success has been
achieved in the field of developing models of social
and political processes (Plotnitskiy, 2001).
The already available models can be tentatively
subdivided into three groups:
1) models-concepts based on discovering and
analyzing general historic laws and representing them
in the form of cognitive schemes describing logical
relations between various factors affecting historical
processes (G. Goldstain, I.Wallerstain, L.N.
Goumilyov, N.S. Rozov and others). Such models
feature a high degree of generalization, but are of a
purely logical, conceptual, rather than mathematical,
nature;
2) special mathematical models of the simulation
type, designed for describing particular historical
events and phenomena (Y.N. Pavlovskiy, L.I.
Borodkin, D Meadows, J. Forrester and others). In
such models, the main attention is paid to carefully
accounting for and describing factors and processes
affecting the studied phenomena. The applicability of
such models is, as a rule, limited by a fairly short
spatial and time interval; they are ‘tied up” to a
particular historical event and cannot be extrapolated
onto longer periods of time;
3) mathematical models, intermediate between the
two mentioned types. These models describe a certain
class of social processes without giving a detailed
description of the features characteristic for each
specific historical event. They are designed for
discovering basic laws characterizing the processes of
the type in question. Accordingly, such mathematical
models are called basic (Malkov, 2004).
In the classical models , the dynamics of nonlinear
systems is modeled using multidimensional
differential equations, difference equations, the
mathematical apparatus of cellular automation, the
mathematical apparatus of the catastrophe theory, the
mathematical apparatus of the self-organized
criticality theory, stochastic differential equations of
Langevin and Ito-Stratonovich, analysis of systems
with chaos and reconstruction of stable states
(attractors) along time series (Malkov, 2004; Haken,
1985; Ebeding, 1979).
Holyst J.A., Kacperski K., Schweiter F. presented
an effective model of social opinion, based on
representing the interaction between individuals in
the form of Brownian motion (Holyst, 2000).
There are also other numerous studies in the field
of modeling social and political processes published
by K. Troitzsch, R. Hegselmann, P. de Vries, D.
Gernert, A. Nowak, R. Vallacher and E. Burnstein, H.
Ader and I. Bramsen, Y.-F. Yung, W. Chan and P.
Bentler, R. Geuze, R. van Ouwerkerk and L. Mulder,
A. Klovdahl and many others (Mikhailov, 2012;
Gutz, 2000).
3 THE MAIN CONCEPTS OF A
SOCIAL CONFLICT
The contemporary literature on sociology abounds in
classifications of types of conflicts according to
various grounds. Consider some of them from the
viewpoint of defining a social conflict as a
mathematical notion in our model.
From the viewpoint of subjects involved in a
conflict, four types of conflicts can be discerned:
1) intrapersonal conflict (it can appear in the
following forms: a role conflict – it appears when
contradictory requirements are imposed on a person,
regarding what the result of his/her work has to be;
intrapersonal – it can result from a mismatch between
the working requirements and the individual’s needs
and values);
2) interpersonal conflict (it can be manifested in the
form of a clash of individuals having different
characters, views or values, and is the most common
one);
3) conflict between an individual and a group (when
an individual assumes a position differing from the
position of the group);
4) intergroup conflict.
Conflicts can also be classified according to the
spheres of activity as: political, social-economic,
national-ethnic and others (Malkov, 2004).
Mathematical Modeling of the Ethno-social Conflicts by Non-linear Dynamics
181
There are quite a few concepts of the theory of
social conflict. Some of the best-known of them are:
L. Coser’s concepts:
• in any society there exists inevitable inequality,
permanent psychological discontent of its members,
interpersonal and intergroup tension (emotional,
psychic disorder), leading to social conflict;
• social conflict as incongruity between the reality
and ideas of various social groups or individuals
about what it should be like;
• social conflict as struggle for values and
pretensions to a certain status, power and resources,
in which the antagonists aim at neutralizing,
damaging or eliminating the opponent (Coser, 2000).
Conflict model of society by R. Darendorf:
• permanent social fluctuations in society, suffering
social conflict;
• any society is based on making some of its
members obey other members = inequality of social
positions in the distribution of power;
• difference in the social position of various social
groups and individuals leading to reciprocal tensions
and contradictions resulting in the alteration of the
social structure of the society (Darendorf, 1994);.
General theory of conflict by K. Boulding:
• all conflicts have common development patterns;
their detailed study and analysis makes it possible to
develop a generalized theory – “the general theory of
conflict” which will allow society to control conflicts,
manage them and predict their consequences;
• Boulding argues that conflict is an intrinsic part of
social life (striving for struggling with the similar is
in the human nature);
• aconflictisasituationinwhicheachofthe
partiestriestoadoptanattitudewhichis
incompatibleandcontraryinrespecttothe
interestoftheotherparty;
• two aspects of social conflict: static and dynamic:
The static aspect is the analysis of the parties
(subjects) involved in the conflict (individuals,
organizations, groups) and relations between them
(classification: ethnic, confessional, professional).
The dynamic aspect studies interests of the parties as
stimuli for conflictive behavior of people. The
definition of the conflict dynamics is a set of
responses to external stimuli (Boulding, 1969).
From the above said, the following important for our
model conclusions can be drawn:
1. A large social conflict is initiated mainly by an
informational and social distance between individuals
or groups of individuals. A basis for such a distance
can root in ethnic, cultural, confessional, as well as
economic dissimilarities.
2. This distance increases in the process of conflict,
especially in its extremal forms (revolutions, civil
wars etc.), bringing the opposing parties to the
attitude of irreconcilability. Unfortunately, history
knows very few examples of short- and medium-term
positive scenarios for such situations.
3. Hence, the point of no return in question is
somewhere before the initiation of conflict, and this
transition of a social system from one state to another
is determining.
4 MATHEMATICAL MODEL
For mathematical modeling, an important point is that
social and political processes cannot be rigorously
assigned. They tend to be subjected to minor changes
and fluctuations. Using analogy, a social process is
similar to a Brownian particle, i.e., a particle moving
along a fairly definite trajectory which, on closer
examination, is highly winding and broken. These
small fluctuations are explained by chaotic motion of
other molecules. In social processes, fluctuations can
be assumed as manifestations of free will of its
individual participants, as well as other random
manifestations of the external medium (Gutz, 2000).
In physics, such processes are generally described
using Langevin stochastic diffusion equation, which
is also, to a certain degree, tested for modeling some
social processes. For example, Holyst J.A., Kacperski
K. and Schweiter F. developed a model of social
opinion (Holyst, 2000).
The model is based on the idea that individuals of
a society interact by means of a communication field
(similar to (Holyst, 2000)). This field is induced by
each individual of the society, modeling
informational interaction between individuals.
However, it should be kept in mind that society,
which is considered here, can hardly be viewed as an
object in classical physical spatial topology. Really,
in terms of transfer of information from individual to
individual, space in society has both classical spatial
coordinates and some additional specific
characteristics. It is because of the fact that in the
contemporary informational world it is not necessary
to be near the object to transfer information to
him/her.
Thus, society is a multidimensional, social-
physical space reflecting a possibility of one
individual to “reach” another individual with his/her
communication field, that is, to affect him/her, his/her
parameters and possibility to move in this space.
Accordingly, the position of an individual relative
SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
182
other individuals also models the level of relations
between them and their involvement in the
informational exchange. Close positions in this
models show that there is regular exchange of
information and social connection between them.
With such a formulation of the problem, a conflict can
be considered to be a type of interaction between
individuals or groups thereof, which results in a sharp
increase of the distance (i.e., social distance – ∆x = x
i
- x
j
, where x is a coordinate in social-physical space,
i, j= [1, N], where N is number of individuals or
consolidated groups of individuals) between them,
and a further increase of the distance testifies to
increasing conflict.
Thus, a communication field can be represented
with a diffusion equation as follows:
,
,
,
̅
,
,
,
,
(1)
In which a divergent type of diffusion is assigned, and
function
,
1
√
,
Is used instead of delta-function; for →0 it
asymptotically tends to the latter, considerably
simplifying the process of computer modeling.
Function
,
characterizes relation between
individuals, which is modeled here using classical
Gaussian distribution
,
1
√
,
,
fairly widely used in various sociological studies.
is coefficient of scientific-technological progress
and development of the i-th individual/group of
individuals.
is coefficient of social activity of the i-th
individual/group of individual.
̅
is inverse Kronecker delta.
Coefficients k
s
and k
c
are used for each separate
individual or group in the system, and a total
coefficient of the entire system is found by fractal
transformation of their values of all individuals and
clusters of the system (Petukhov, 2015a; Petukhov,
2015b; Petukhov, 2016a; Petukhov, 2016b).
Translations of an individual are described using
Langevin equation:
,
,
√
2
,
(2)
where stochastic force
is introduced, which
models a random factor in society, and, in particular
cases, external effects on individuals.
When solving equations (1) and (2), differential
equation
should also be taken into account.
In a general case, initial conditions for equations
(1) - (3) can be taken as follows:
|
,0
.
It is also necessary to assign a range of characteristic
parameters 0
,
,1 (individual
distribution).
5 APPROXIMATE SOLUTION OF
THE SYSTEM
For a simplest model of two interacting individuals or
two consolidated groups of individuals (i.e.,
belonging to the same social, confessional, ethnic etc.
group), assumed to be in a state of conflict,
accounting for external effects, equations (1) and (2)
can be written as:
if
112 2
112 2
,
1
,,
csc s
kkk k
csc s
kkkk
then
2
2
12
2
2
2
12
2
1
11
1
21
2
22
1
12
2
11
1
1
2
1
22
2
2
1
,
,,0
,
,
,,0
,
,
2,
,
2,
xx
cs
xx
cs
cs
cs
hxt
Dhxt hx
t
kke
hx t
Dhx t hx
t
kk e
hx t
dx
kk D t
dt x
hxt
dx
kk D t
dt x
(3)
Mathematical Modeling of the Ethno-social Conflicts by Non-linear Dynamics
183
To obtain approximate analytical solutions of system
of equations (3), series expansion is used with the
accuracy of up to the quantity of the first order of
smallness for ∆
,∆0 of
difference
,
,
∆
∆,
(4)
Then, assuming the following initial conditions:
0,
(5)
,0
1,
the first two equations of system (3) are integrated,
using (4), (5), after which the following expression
results:
2
2
2
0
1
2
0
,
,
2
3.
ij
t
ii
t
xu x u
ji
cs
hxt D x udu
t
D
kk e du
ji
(6)
Using expression (6), the last two equations of system
(3) can be transformed, based on the continuity of all
the functions, into the following form:
1
2
11 12
1
12
2
0
1
1
2
22 21
2
21
2
0
2
2
2
12
2
2
2
12
2
1
2
2,
1
2
2.
xu x u
t
cs cs
xu x u
t
cs cs
dx
kk Dt kk x u x u e du
dt
Dt
dx
kk Dt kk x u x u e du
dt
Dt
(7)
After time differentiation of (7), the following forms
of differential equations are obtained:
2
2
12
2
2
2
12
2
2
11
1
2
1
21112
12
2
1
2
22
2
2
1
22221
21
2
2
21
2,
.
21
2
cs
xx
cscs
cs
xx
cscs
dx
kkD
dt
kkkk
xxe
dt
D
dt
dx
kkD
dt
kkkk
xxe
dt
D
dt
(8)
To further simplify the solution of the problem in
question, it is assumed that equality of active
stochastic forces for individuals or various groups
12
tt
is satisfied.
Then, introducing new designations:
12
11 2 2
2
1112 2 2 21
2
2
2
,
,
1
2,
1
,
cs c s
cscs cscs
yx x
ADkk kk
B
kkkk kkkk
C
after finding the difference of equations (8), the
following equation is obtained:
2
2
2
,0,0.
Cy
dy
ABye B C
dt
(9)
Now, equation (9) is rewritten in Cauchy form:
2
,
.
Cy
dy
z
dt
dz
ABye
dt
(10)
System (10) can be viewed as a dynamic system
describing a process of interaction between two
individuals or groups thereof.
As is known (see (Goryachenko, 2001; Andronov,
1981)), a dynamic system describes a process of
transition from one state to another. The phase picture
SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
184
of system (10) will be represented by a set of all
states; to determine equilibrium states of this set, it is
necessary to solve the following system of equations:
2
0,
.
Cy
z
A
ye
B
(11)
Analysis of system (11) is readily represented
graphically (Fig. 1):
Figure 1.
As follows from Fig. 1, two equilibrium
conditions are possible if condition
0,
2
1
2
1
2
1
2
1
B
A
e
СB
A
e
С
,
(12)
is satisfied, and one equilibrium condition if one of
the three following equalities are satisfied:
0
B
A
,
2
1
2
1
e
СB
A
,
2
1
2
1
e
СB
A
(13)
Naturally, if conditions opposite to (12) are satisfied
2
1
2
1
2
1
,
2
1
e
СB
A
e
СB
A
, A/ B
≠0
(14)
there are no equilibrium conditions.
As system (10) is conservative, the law of
conservation of energy holds. Then, knowing the
energy integral of system (10), it is possible to find
phase trajectories of the system in question. As is
known [10], in a conservative system, phase
trajectories are lines of the level of the potential
energy function, which has the following form:
2
2
0
2
y
Cy
Cu
Be
VABueduAy
C
(15)
For social systems the notion of energy is either
meaningless or has another definition. However, their
dynamic behavior qualitatively coincides with the
behavior of conservative mechanical systems, and, in
the phase plane, the qualitative behavior of their
phase trajectories is similar (Goryachenko, 2001).
As among parameters
CBA ,,
only
A
can
invert its sign, only two possible situations are to be
considered. First, if conditions
0,0
2
1
2
1
A
B
A
e
С
(16)
are satisfied, relations V(y) represented in Fig.2 and
the related phase trajectories are realized, where
.
Second, if conditions
0,
2
1
0
2
1
Ae
СB
A
,
(17)
are satisfied, relation V(y) represented in Fig.3 and
the related phase trajectories are realized.
Based on the analysis of the obtained phase
pictures (Fig.2 and Fig.3), it can be concluded that
there exists a certain stability region (confined in the
pictures by a stable loop of the separatrix), i.е. the
region within the closed trajectory.
Figure 2.
Mathematical Modeling of the Ethno-social Conflicts by Non-linear Dynamics
185
Figure 3.
The boundaries of this region are defined by
values of characteristic parameters of individuals, or
groups thereof, as well as the society as a whole:
,
,. These coefficients, strictly speaking, can
change in time as a result of the interaction of
individuals, thus, affecting the dimensions and
position of the stability region. However, in the
present study, only a short-term scenario was
considered, thus, their possible variation in time was
assumed insignificant.
Individuals and groups thereof having parameters
necessary for getting into the stability region at an
initial time do not move apart from one another to a
relatively large social distance as a result of reciprocal
interaction. They remain at a distance within which
social relations and active informational exchange are
possible.
This can be interpreted as existence of an
interaction region, parametrization of which makes
relatively abrupt fluctuations of social coordinates,
i.e., a state of conflict, highly improbable or
impossible.
It is true that in a society, where social and
informational contact, mutual permeation of different
cultures and ethnic groups is sufficient, where
particular groups of population do not separate
themselves, creating closed subsystems (in which
conditions substantially differ from those of the main
system), the possibility of initiating ethnic-social,
confessional etc. conflicts is relatively minimized.
Outside the stability region, phase trajectories are
divergent and not close. Individuals/ groups of
individuals that are, at an initial time, outside this
region, after some time will find themselves at a
relatively large social distance, which corresponds to
the increase of social and informational gap between
individuals and/or groups of individuals. It is this
state of a social system that can be characterized as a
conflict and manifestation of the contradictions
existing between individuals and groups thereof.
With the introduction of social friction in the system,
the picture changes significantly – Fig.4.
Figure 4.
However, it is a separate task, and it requires a
separate article.
Thus, in ethnic-social conflicts, this is manifested
in the minimization of social and cultural contacts
between different ethnic groups, growth of the social-
economic gap, aggravation of contradictions and, as
a result, transfer to the phase of explicit confrontation
accompanied with the destabilization of the social and
political system as a whole.
6 CONCLUSIONS
Social hyper-clustering of a society, abrupt
distinctions in the informational and social
environment of individuals, cultural and inter-ethnic
separation create ideal conditions for social conflict.
Therefore, prevention of conflicts in society,
determination of boundary conditions of their
initiation and search for the most effective scenarios
of their suppression is a vital issue for contemporary
social sciences.
The present article concisely reviews the main
approaches to modeling in social sciences, problems
of determining social conflict and its main concepts.
Conflict in a social system is defined in terms of
mathematical modeling.
A mathematical model based on Langevin
equation is introduced, an analytical solution in the
first approximation for the divergent type of diffusion
is given.
SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications
186
It is shown that, even in a simple example of two
interacting ethno-groups of individuals, the
developed model makes it possible do discover
characteristic properties of conflict in a social system,
to determine the impact of social distance in society
upon the conditions of generation of such processes,
accounting for external effects and random factor.
As a result of the modeling, a certain region of
stability for a social system is found, within which it
is stable and conflict-resistant.
The results of the present investigation will make
it possible in future to approach the analysis of
general problems for large numbers of individuals.
ACKNOWLEDGEMENTS
The research work was financed from the Grant of
Russian Scientific Fund (Project №15-18-00047).
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