=Paper=
{{Paper
|id=Vol-3200/paper4
|storemode=property
|title=Method of Calculation of Information Protection from Clusterization Ratio in Social Networks
|pdfUrl=https://ceur-ws.org/Vol-3200/paper4.pdf
|volume=Vol-3200
|authors=Vitalii Savchenko,Volodymyr Akhramovych,Oleksander Matsko,Ivan Havryliuk
}}
==Method of Calculation of Information Protection from Clusterization Ratio in Social Networks==
https://ceur-ws.org/Vol-3200/paper4.pdf
Method of Calculation of Information Protection from
Clusterization Ratio in Social Networks
Vitalii Savchenko 1, Volodymyr Akhramovych 2, Oleksander Matsko3 and Ivan Havryliuk 4
1,2
State University of Telecommunications, Solomianska str.7, Kyiv, 03110, Ukraine
3,4
The National Defense University of Ukraine named after Ivan Cherniakhovskyi, Povitroflotsky av. 28, Kyiv,
03049, Ukraine
Abstract
The article investigates the dynamic models of the information protection system in social
networks taking into account the clustering coefficient, and also analyzes the stability of the
protection system. In graph theory, the clustering factor is a measure of the degree to which
nodes in a graph tend to group together. The available data suggest that in most real networks,
and in particular in social networks, nodes tend to form closely related groups with a relatively
high density of connections; this probability is greater than the average probability of a random
connection between two nodes.
There are two variants of this term: global and local. The global version was created for a
general idea of network clustering, while the local one describes the nesting of individual nodes.
There is a practical interest in studying the behavior of the system of protection of social
networks from the value of the clustering factor. Dynamic systems of information protection in
social networks in the mathematical sense of this term are considered. A dynamic system is
understood as any object or process for which the concept of state as a set of some quantities at
a given moment of time is unambiguously defined and a given law is described that describes
the change (evolution) of the initial state over time. This law allows the initial state to predict
the future state of a dynamic system. It is called the law of evolution.
The study is based on the nonlinearity of the social network protection system. To solve the
system of nonlinear equations used: the method of exceptions, the joint solution of the
corresponding homogeneous characteristic equation. Since the differential of the protection
function has a positive value in some data domains (the requirement of Lyapunov's theorem for
this domain is not fulfilled), an additional study of the stability of the protection system within
the operating parameters is required. Phase portraits of the data protection system in MatLab /
Multisim are determined, which indicate the stability of the protection system in the operating
range of parameters even at the maximum value of influences.
Keywords 1
dynamic models, information protection system, social networks, clustering coefficient,
nonlinearity, exception method, homogeneous characteristic equation, function differential,
system stability, phase portrait
1. Introduction also various: with the help of differential
equations, discrete mappings, graph theory,
Markov chain theory, and so on. The choice of
Descriptions of dynamical systems for various
one of the methods of description determines the
problems depending on the law of evolution are
III International Scientific And Practical Conference “Information
Security And Information Technologies”, September 13–19, 2021,
Odesa, Ukraine
EMAIL: savitan@ukr.net; 12z@ukr.net; macko2006@ukr.net;
ivan.havryliuk@gmail.com
ORCID: 0000-0002-3014-131X (A.1); 0000-0000-0002-6174-
5300 (A. 2); 0000-0003-3415-3358 (A. 3); 0000-0002-3514-0738
(A. 4)
© 2021 Copyright for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
specific form of the mathematical model of the coefficient for complex networks. In [14], it was
corresponding dynamic system [3]. concluded that based on the results of the
The mathematical model of a dynamic system experiment, it can be concluded that among the
is considered to be given if the parameters clustering algorithms there is no universal
(coordinates) of the system are introduced, which algorithm that would be significantly ahead of
unambiguously determine its state, and the law of others on all data sets. The leaders of
evolution is specified. Depending on the degree of benchmarking are the algorithms Spinglass and
approximation to the same system, different Walktrap. From the considered analysis of the
mathematical models can be matched. works, it can be concluded that currently the
Theoretical study of the dynamic behavior of a protection of users in social networks is
real object requires the creation of its considered primarily as a technical problem that
mathematical model. In many cases, the does not take into account the structural
procedure for developing a model is to compile parameters of the network and its topological
mathematical equations based on physical laws. features. This emphasizes the relevance of the
Usually these laws are formulated in the language topic of work regarding the construction of a
of differential equations. As a result, the protection system based on structural parameters,
coordinates of the state of the system and its taking into account network clustering.
parameters are interconnected, which allows us to
begin to solve differential equations under 3. Formulation of the research task
different initial conditions and parameters.
It is necessary to investigate the dynamic
2. Related works system of information protection in the social
network (SN) from the clustering factor. Carry out
In the article [1] the definition of the clustering modeling of a nonlinear protection system taking
coefficient in the case of (binary and weighted) into account the clustering factor in SN.
directional networks is extended and the expected Investigate the stability of the protection system
value for random graphs is calculated. In [2], it is in the SN.
noted that the properties of the small world of
neighboring connections are higher than in 4. Main part
comparative random networks. If a node has one
or no neighbors, in such cases the local clustering 4.1. Nonlinear solution of the
is traditionally set to zero, and this value affects protection system in the SN,
the global clustering factor. It is proposed to taking into account the action of
include the coefficient θ for isolated nodes in
order to estimate the clustering coefficient, except a specific parameter ‐ the
in cases from the determination of Watts and clustering factor
Strogats. In [3] a method of determining trust and
protection of personal data in social networks was Analysis of graphical dependences of a linear
developed. In article [4-6] the clustering system [3] indicates the nonlinearity of the
coefficients for social networks, including power system. Therefore, in the system of equations (1)
ones, are considered. In [7], a comparison of we introduce nonlinear components (2):
different generalizations of the clustering
coefficient and local efficiency for weighted
dI
undirected graphs is made. In the article [8] the Z p Z ( Cv С K ) I
analysis of the clustering coefficient on the social dt
C
network twitter is carried out. In [9], an analysis dZ ( vV v 1 ) I ( C (1)
dt d 2 Cd 1 )
of the clustering coefficient through triads of N2
connections was performed. In the article [10] the
dependence between the clustering coefficient
where: Cv1 - the total number of connections
and the average path length in a social network is vV
investigated. In [11,13] the use of clustering in the network, N - the number of vertices in the
methods of social networks for personalization of network.
educational content is investigated. The article
[12,15] discusses the behavior of the clustering
We obtain a system of equations:
dI Z Z( C С ) I L ( I 2Sin2t )
dt p v K 2 0
3 3
L3 ( I0 Sin t )...
Cv1
dZ
(
v V )I ( Cd 2Cd1)К2 ( Z02Sin2t )
dt N 2 (3)
K ( Z 3Sin3t )...
3 0
Figure 1: Differential of the clustering coefficient
function Let's rewrite the system and present it as
follows:
dI 2 3 dI Z I L I k sin k t ,
dt Z pZ (Cv СK ) I L2 ( I )L3 ( I )... dt 1 k 0
k 2
Cv1
(2) dZ
dZ ( vV )I (Cd 2 Cd1)К2 ( Z 2 )K3 ( Z3 )... k k (4)
dt 2 dt 2 I K k Z 0 sin t ,
N k 2
where: L2 , L3 , etc. K 2 , K3 , etc. some linear where:
operators. We consider the nonlinearity of the Cv1
system to be weak, which allows us to find a
solution for each equation of the system (2) by the
Z p , 1 Cv CK , 2 Cd 2 Cd1 , (
vV
N2
)
method of successive approximation, putting: Next, use the exception method:
I I1 I 2 I3 ...
dZ
2 I K k Z0k sin k t
dt k 2
1 dZ
I K k Z0k sin k t
2 dt
k 2
dI 1 d 2 Z 1
2 kKk Z0k sink 1 t cost (5)
dt 2 dt k 2
Figure 2: Differential of protection function
Since the differential of the protection function
has a positive value in some data domains (the
requirement of Lyapunov's theorem for this
domain is not fulfilled), an additional study of the
stability of the protection system within the
operating parameters is required
Z Z1 Z2 Z3 ...
Let at
dI dZ
dI 0 , 0, and dZ 0 , 0
dt dt
I I 0 Sin t , Z Z0 Sin t
Figure3: Graphs by dependence (4)
Figure 4: Graphs by dependence (5)
Substitute all the found expressions (5) in the
first equation of system (4):
1 d2Z 1
2 dt2
k2
kKkZ0k sink1t cost
1 dZ
Z K Zk sink t
2 dt k 0
k2
Lk I 0k sin k t (6)
k 2
or:
d 2Z dZ
2 1 2Z
dt dt
1
kK Z k sink 1t cost
k 2 k 0
1 1 Kk Z0k sink t
k 2
2 Lk I 0k sin k t
k 2 (7)
Figure 5: Graphs by dependence (7)
Now we find a common solution of the
corresponding homogeneous equation:
Z 1Z 2 Z 0 (8)
The characteristic equation has the form:
2 1 2 0 . Consider the case of the
positive discriminant of this equation:
2 1 12 4 2
D 1 4 2 0 1,2 (9)
2
From:
1 12 4 2
t
1 12 4 2
t
Given (13,14,15) we have:
Z одн (t ) c1e 2
c2 e 2
1 1242
1 12 42 t
joint solution of a homogeneous equation. t e 2
Z(t) (N(t) e 2
)dt
To find the general solution of the 12 42
inhomogeneous equation we use the method of
1 12 42
variation of arbitrary constants: 1 12 42 t
t e 2
N(t) e 2
)dt,
1 12 42 1 12 42 t12 42
t t (16)
Zодн (t ) c1 (t )e 2
c2 (t )e 2
where: c1 (t ), c2 (t ) are from the system:
1 12 42 1 12 42
t t
c1 (t )e 2 c2 (t )e 2 0,
2
1 1 42
1 12 42 t
c1 (t ) e 2
2
1 12 42
2
1 1 42
t
c ( t ) e 2 N(t ),
2 2
where:
(11)
From equations (10, 11) we obtain:
1 12 4 2 1 12 4 2
t t
c1 (t )e 2
c2 (t )e 2
Figure 6: Graphs by dependence (16)
1 1242 24
t 1 1 2
2 2 N(t )
c2 (t)e 3.2. Define the phase portrait of the
24 (12)
1 1 2
2
data protection system
or:
1 12 42
Initial equation:
t
c2 (t)e 2
42 N(t)
1
2
(13)
d2Z 1
where will we get: dt2
dZ
1 dt 2 Z
k 2 k 0
kK Zk sink1t cost
(17)
1 12 42 1 1 Kk Z0k sink t 2 Lk I0k sink t
1 t k2 k2
c2 (t)
4
2 N(t)e 2
dt (14)
1 2
The solution will be implemented in the
1 12 42
1 t program MatLab / Multisim. Let's make the
c1 (t )
12 42
N (t )e 2
dt (15)
scheme (Fig. 7).
The phase portrait is presented in the form of The results of the program are presented in
an ellipse, which indicates the stability of the Fig. 8, 9.
personal data protection system.
Figure 7: Block diagram of the phase portrait program in the Multisim program, taking into account
the attack block
Figure 10: Harmonic oscillations of the
Figure 8: Harmonic oscillations of the protection protection system on time Z=f(t) taking into
system on time Z=f(t) account the attacks
Figure 9: Phase portrait of the protection system
on clustering factor
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Akhramovych, V. Samosyuk, G. Shuklin,
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